Hi viewer- this is easy-healing channel, i’m vincent kononwa and i’ll be taking you through from two mathematics indices and logarithm stay tuned.
As i welcome you all, i shall start with indices and analysis? We shall consider certain numbers, such as i’ll consider number 2 5, which is the short form of writing 2 multiplied by 2 up to five times. This is short formed to this now here, as you can see, these two is referred to as the base and five here is the index or the power. Now, let’s just refer to his first as the power to which the best two is first two. Now this five here we have say is the part which two is rest to mean that these two has to be multiplied itself up to five times now five year or the number to which the best is raised is referred to as index, which is in plural, referring to us indices. Now we shall consider in general form in case we have a raise to n.
This simply means that air is the best, but we have to multiply a up to n times, whereby n here is a positive, integer or maybe a negative, but it is an integer a number?
So a is raised up to n times, meaning that a is the base and n is the index to which the base has to be raised upon now we shall see some of the examples.
For example, we may see like a cube having a number like a cube.
This simply means that we have to multiply a three times so meaning that a cubed is the short form of writing a multiplied by a multiplied by a and if we have another number like 4 raised to 4.
This 4 here is the index to which the base 4 is raised, i’m referring to the base 4 and for there being the index or the power! Now this simply means we have to multiply 4 up to 4 times, and if you have that one, you will have to get the value as the first one will give you 16 64, multiplied by four! You will get 256 so meaning that 4 raised to 4 is giving you an answer of 256. So writing this one can be short, formed or short written into 4 raised to 4. .
Now this is what we are referring to as indices, so meaning that index is the power to which any number is raised upon.
Next, we are going to see the laws of indices. Let’s now handle loss of indices under laws of indices?
We shall consider several kinds of numbers.
For example, let’s consider the following: we have f5 multiplied by a squared raised to 5 by s squared here you will identify that you have to multiply the first one will be a multiplied by a five times.
Then you continue multiplying a twice, so we shall continue twice this one here represent the last a squared, and this f raised to 5 is the short form of writing a up to this point. This one here is what we have short formed to a raised to 5, and this one is what has been transformed to a square having that, as we had said earlier, a here is the base and 5 is the index to which a is written. Now, if you do this, you will end up having this one will give you a base, since the base is the same, it does just means that we are going to multiply all of this, and we shall get a raise to 70, because you will have seven of them seven times of a’s, so it will be a raise to seven in otherwise this one can be short formed or should written as instead of writing a long method or a long way. You can still write a 5 raised to 5. Then you add the powers, and you will find the same answer in that this one will give you a raise to 7 this one here will give us the first law of in the indi in this indices?
It will give us the first law of indices and that law is called a sum of two powers, because we have dealt with the two powers of indices, so it will be. The sum of two is referred to as the sum of two powers. Now, instead of just going the long method, you can just do the short form of it.
Uh whereby you add the power since the base are the same, and you get the value there. Let’s consider another one with the division so that we may have a rest! For example, let’s say 5 divided by a raised to 2.
This is the quotient rule, so we shall apply the quotient rule here and we shall have a is the same as writing a raised to 5? Divided by a squared, and here, if we write it in long method, we shall find that this one is the same as writing a multiply a five times by a multiplied by a? If you have that, since the items there are identical, we can simplify it by simplifying two of them on the numerator, which can say denominator by getting one here, and you will find the remaining will be a multiplied by a multiplied by a and you will have a cube, meaning that if you have a raise to 5 divided by a square on simplification, you will get a cube in. Otherwise you can still work out in a short form of this, whereby we have a raise to 5 divided by a squared, and this one you can work out by taking? Since the base is the same, it only works if the base is the same since the base here is the same, you can take a raise to 5 minus 2 that one here will still give you the same answer as a cube, and this one here is called the difference! Difference of two’s powers that one there is the difference of two powers whereby we have dealt with the first power of five and that of square, but sometimes you may have numerous of the powers, but with the same bases you will still do the same, but the difference of the powers will depend on how many items are you dealing with for a for example? maybe this one could be included with another division here and could be divided by 2 a raised to 2. The same could be applied whereby now here you could include minus 2 so as to have a raised to 1. . In case of that, we shall say that is a difference of three powers! But for our case this one was an exclude and we got a raise to three. We shall consider another one whereby we have a raise to five, but this one is still in a bracket or is bracketed and then two placed outside.
In that case, this just means you have to take a multiply by a multiplied by a five times. Then, whatever you have here, that is the first, which is in bracket, but it is being squared? If you write in long method, the first one being squared means you have to take the same same value here. You multiply itself twice so if this one is five times, you continue multiplying by the same base. In that you will have this one is a raised to five the same to the one which we have. Otherwise, what is in the first bracket is a raised to 5 and what is in the second bracket is a raised to 5. . So if you do the same, you will find that if you multiply this and you open the bracket and multiply again with the second bracket, you will end up having a raise to 10! You will end up with a raise to 10, so meaning that we can just simply take. If you have a raise to 5 squared, you can just take a since the a is the same best.
Then this 5 here we just multiply the powers if it is in bracket and the adjacent bracket is being squared, no matter what power will be outside or inside what you just do you multiply the powers and you will get this one is a raise to 10 the same value which we have obtained here, the same answer if you work it in a long method, as we have seen this one here is known as the product of two squares that is known as the product of two squares with that we can give a general form of it whereby we shall say in case we have a raised to m? Let’s take a b raised to power m, be multiplied by a raise to n.
Then this one! According to the first law of index, which we have observed, and that is the sum of two powers, we should have a raised to m plus n. That is what we obtain when we work the first example using a as the base and the power five and two. We found that this is the first law of in indices, and we have say this is the law of the sum of two powers. Then another law we observed when we did with a division or quotient, and we said in case we have a raise to m being divided by a to power.
N we shall find is the same as taking a raise to m over a n, but this one can be simply be worked out by a raised to m minus n, and this is the value for what you will find after working it out! Then the third law we have observed, which is now the product of two sum, the product of two powers, and we have found that this one can be given by. If you have a raise to m and outside of the bracket we have n, you can still find the value of that by taking a. But now the powers has to be multiplied, so you will have m multiplied by n. So this is the value which you will find if you multiply it, and this one is the third law of indices. Next, we are going to see some of the examples if we work it out with the same bases. Let’s take the first example containing the first rule or pertaining the first rule of index or the law of indices. So we shall consider an example whereby we have 5 raised to 7 b, multiplied by 5 cubed and according to the first law of index, we shall find out that this one is the same as taking five. Since five is the same base to the two items there. We shall take five then to the powers we shall add, because it’s the first law, so we shall add the past and you will find it is giving you five raised to 10 and if you do the long way of it or the long form you will find you still arrive at the same answer in case you have 6 raised to 9 being divided by 6, raised to let’s say 4. You will find out this one lies under the quotient rule, and that is the difference of two powers and if you work it out, you will find? Since you have the same best, you will end up having nine minus four. As per the rule.
The second rule of index says so you will end up with 6 raised to 5, and that is the value of that calculation. And if you work a long way of it, you will still arrive the same value! Let’s see the third example and this one pertains the product of two sums at the product of two powers, and this one is for example, if you have three squared but being raised to four, you will find we say this one. Since we have the same best, we shall just take three but the powers we shall multiply and we shall find this! One is going to give us three raised to eight, and that is some of the examples you will have you will find or encounter during when you work out some of the problems pertaining loss of indices. Next, we are going to see how we can evaluate the zero index, and what do we mean by zero index? now now, let’s consider the zero index and we shall consider an expression such as a raised to m being divided by a raised to m whereby they have the same base, but we are dividing according to the second law of indices. We say this one can simply be worked out by taking a raise to m minus m or you can still do it the long way by taking a raise to m, then you divide by a raised to m, and you will find that here you are going to divide and you will find here a raised to m into a raised to m you’ll find is one and the same applies to here, so you will end up with the others one. But if you do it using the second law of indices, you will have this one as a raise to m minus m, you will get 0, so it will be 0! So this simply implies that in case you have a number being raised to m or to a given power. Then you divide the same number or the same base b raised to the same power? You will end up having that number being raised to zero and any number be it a thousand or a million being this to zero. It will give you one, so we can say that if you have a number a raise to zero, you will find that the value of the result will be equal to 1! That implies that, no matter how big the number is, for example, if we have that you raise it to zero, you will still pertain the same value as one or, if you have, for example, let’s take five raised to four, then you divide by 5 raised to 4? You will end up having 5 raised to 4 over 5 raised to 4, and this one will give you 1 or if you do it, using the the law of indices. You will find this? One will give you 5 raised to 4 minus 4, and you will get this. One is 5 raised to 0 and we have said any number being raised to zero will give you one.
So that is what we are referring to zero index. So here this zero here being raised to is what we are referring to zero index, and we have seen that if you raise any given number to zero, you will get one. So next we are going to view some of the examples on what we have learned there and how we can find some of them down.
Let’s consider an example such as you have been told to solve 2 into x, equals 2 to 12, and another thing you should pertain in your calculation before you do any exercise on indices is make sure that whatever you work with is of the same identity or base. So if you want to work any number with the same index or to a given index, make sure if it includes the product rule sum or that difference rule, then you have to make sure that the base is the same, for instance, for this our case here, it simply means that if we take 2, then we use this product. Rule should be equal to 2 raised to 12, so meaning that our 2 will be x, multiplied by 3 and we shall get 3x should be equal to 2 raised to 12, and if you do that, it simply means that, since the bases are the same, then the powers also are the same. So we can take the powers a lot. So as we can find the value of x and to find the value of x, you have to divide both sides by 3, and you will find your x is equal to 4, so meaning the missing value there in the above expression is 4. . So if you replace here 4, you will find that the two sides are equal. Next, i will give you one to work on it? Uh, whereby you have this?
One is an exercise for you. We have three two x raised to four is equal to eight one, and it is that make sure if you’re dealing with the indexes make sure the bases are the same!
So if that is the case, i believe you can work out that one by using the same procedure as we have done there. Let’s, let’s see how we can find negative indices, or what do we mean by negative indices, and we shall consider a given expression such that we have a raise to negative n, which is a negative index being multiplied by a raised to n. According to the first law of indices, we shall find out that if we use that we are going to have a raised to negative n! But since we have say the first law and first law is the sum of two powers, then we are going to add the power, since the bases are the same, and if we do that, we are going to have a raise to 0. So that implies that a raised to negative n multiplied by a raised to n is equal to a raised to 0. And if that is the case, we can say that if you have a raised, negative n is the same as writing a raise to 0 over a raised to a raised to n?
That means we are going to eliminate a raise to l on this side by dividing both parts by a raised to n, in that, if we do that way, we will have this kind that this side, we are dividing by a raise to n a raised to n, so that we may remain with the a raised to negative n. And if that is the case, we are going to have a raised. Negative n is equal to a raised to 0 over a raised to n, but remember one thing: remember: we say that a raised to zero is equal to one. That was the zero index that if you have any number being raised to zero, you will get one, and in this case here we have a raise to 0 over a raised to n, so meaning that, in case you have n raised to negative n will be equal to 1, because we are going to replace where we have a raise to 0 by 1 and being divided by a raised to n. So that means that in case you have a number or a negative index, then is the same as writing one over that number without the negative index. So the negative index changes to positive index, as we have seen there. Let’s view an example: for example, we have 2 cubed divided by 2, raised to 4. . If we have a certain number like that, then it means? If we use the second law of indices, then we shall have 2 into 3 minus 4. Since we have the same base, then we have to subtract the powers and we shall end up with 2 raised to 1, and since that is what we have remember, this 3 here is less than 4, so it shall be 2 raised to negative 1. If that is the case, remember what we have said since we have 2 raised to negative 1. We have said according to this law, here is the same as saying: 2 raised to negative 1 is equal to 1. Over now, a n will be the number here, a raised, negative n without the negative sign so meaning they are errors! Negative n is being represented by our number here.
2 raised negative 1. . Now we shall rewrite it with 2 raised to 1, and this one is the same as writing 1 over 2, which is a half, so that is how we do rewrite it. That is the negative index in case! We have another example such as 2 over 3 raised to negative 2. . In case of that, then considering the same law of negative indices, we shall find out that this one here is the same as writing. 1 over this number here, 2 this 2 over 3 is the same as a which is negative? 2 is the same as writing! 2 over 3 is the same as a raised to negative n, whereby negative 2 is what we are have here! Negative n and 2 over 3 is our a so having that we can still rewrite that one in that form and we shall have 1 over 2 over 3, but now the power will be positive. So that is what we shall have and if we have that we shall say it is 1 over now this one will be 4 over 9. But if you take 1 divided by 4 over 9, we are going to have nine over four. You only end up with nine over four? If we work out, you will get that’s what you will have nine over four and that one is going to give you three over three over two? If you simplify squared it’s just getting back to what you have so meaning that in case you have a fraction being raised or yeah being raised to a negative index! For example, here i have 2 over 3 raised to negative 2! The value which i will get is just the same as interchanging the fraction, which will be for this case will be. If i interchange, i will have 3 over 2 squared and if i have 5 over 6 raised to negative 3, then i will just interchange here 6 over 5, then the power shall remain positive 3. That is the value which i will find. We all found that by working it out and we have seen how it works out in case it’s a fraction, but if it is not a fraction, then the laws should be followed correctly, as we have been working it out for your case now, work out and see if you can find out in case, you have been given 6 raised to negative 4 being divided by or divided by, 6 raised to negative 3! Now, let’s just take positive 3!
Work out the value of that and see what you will find bear in mind. The negative incest work at that? Next we are going to see another one and that’s the fractional index. We see how we can evaluate, or what do we mean by fractional fractional index? we shall find that fractional index most of the time are written in the form of p over q, and in this case here, you will find out that p and q are integers, and these integers are in such a way that q is not equal to zero. Q is not equal to zero and they don’t have any other common factor. Apart from having a common factor, one so meaning that p and q, their only common factor is 1? That is how you will find fractional index being written in form, so meaning that in case we have a squared b equal to b. Then this one here it simply mean that a here is equal to square root of b.
How? how is this possible related to fractional index? we have said fractional index is written in the form of p over q, but here i have written square root of b be equal to a whereby s squared is equal to b. It simply means that this one here can be written further by saying that a here can be obtained by taking a, but now this square root. Here is the same as writing! 1 over 2 2 here implies that this i have implies that we have to find the square root of it is b. A half so implies that in case you want a you have to find the square root of b? You can either like rewrite the square root of b in this form or write it in the other form, so meaning that we have m equals to b a half which is simply writing the square root of b. So this is this i have here is what i’m referring to as fractional index! The same applies in case.
We have a cube, you will have. If a cube is equal to b, then it simply implies that in case you want the value of a you have to find the cube root of b and finding the cube root of b is just the same as finding the cube root, but written in fractional form the fractional index form.
You will find that it is b raised to a third, and if you have a raised to 4 b equal to b, then it implies that a here is equal to the fourth root of fourth root is written in that form of b, and this one in fractional index can be rewritten by four r raised to a quarter. So if you find a number being written in that form, note that that one there we are finding the fractional index of the given number in order to find a given, integer or number. So we shall consider an example that such that we have 16 raised to a half, but it’s being squared? Let’s just consider an example?
16 raised to a half being squared and this one here we can see that the one in bracket here it is under the fractional form, and we can see that if you write it in a long way, you will find that you are going to have 16 raised to a half multiplied by 16 raised to a half. Since it is square, it means we are going to rewrite the same number in the bracket itself! Twice- and we shall have that value there and according to the first law of indices, you will find that this one is the same as writing. 16 raised to a half, but since we have the same base, it will be a half plus a half, and this one here is going to give you 16 rest 1, which is the same as 16! Following the fractional index law. You will just find out that 16 raised to a half, which i have said simply is the square root, so you will have the square root of 16 and you will have to raise it to 2 rewriting this one in this form, so 16 raised to half can be written as the square root of 16 and then the square being outside, and if that is the case, you will find that the square root of 16 you will find is equal to 4! But remember this is what we have in bracket and outside the bracket. We have raised to power two and four being squared or four multiply itself twice. You will find your answer being 16, so meaning that 16 raised to a half squared you will find. The answer is 16?
, whether you work it with the other way or you work it directly. As using the fractional index, you will still get the same value so in general! That simply means that in case we have a rest one over n.
It simply means that this one here will be equal to. Since a is the best year and 1 over n is the fractional index, then we can still write that one in a form of whereby a will be in the square root sign, and here what we have here as the fractional index and here will be the root to which a must be found? So here we shall have the root n, so we shall have n root of a and then we can extend by saying that in case we have another one whereby we have a being raised to m over n. We shall rewrite this one? As since this one is a fraction of index, we shall still write this one in form of a root sign whereby m will be inside the root side and n. Here will be the integer or the number to which the number under the root sign must be found. So we shall find it to that one, but after finding it we have to raise it. The numerator of the fraction so will be to power m. With an example, we shall view 25, let’s find 25 3 over 2, so 25 raised to 3 over 2 is the same as rewriting that number in the form of 25 the root side 25, and here we place two, which can be just be simply the square of side and all of it.
We shall find the cube of that value, so meaning that we are going to find 25, but the square root of it. 2 means we are having the square root of 25 and 3. After finding the value you have to find cube root of that value as according to this law, so we shall end up having the square root of 25 is 5, but we shall cube so you will end up having 125 as the value of what we have there, and that is simply what it means?
By having a raised to m over n as a fraction index being written in this form and already calculation, you will find it lies under the same calculation as we have done with that. I believe that you will have to work any form of sum or computation or problem. You may encounter dealing with indices as we have worked out now, let’s see or learn about logarithms, and we shall consider given numbers under logarithm! For example, we are going to have 4 raised to 0 and remember from our indices? We save a 0 index such that 4 raised to 0 will give us 1 and if we have any number raised to 1 is the number itself! But if you square 4, you will get this one as 16 and if you cube 4, you will get 64 and if you take this forest to 4, you will have to get 256! Here. You can continue with the least endless such that you may even add forest to 5 4s to 6 until onwards. But what we are interested in today is local themes, and we shall find we said that here, 0 1, 2, 3, 4 up to the n we said, is referred to as the index or indices for them!
Our numeracy associated b indices now logarithm here just means the indices! 0 1. We have indices, 0, 1, 2, 3 4 up to the endless, are the logarithms of the corresponding numbers and the numbers i’m referring to are 1 4, 16, 64, 256 endless to the basis of 4, so meaning that 0 1 2 3 4 are the logarithms. But for this case you can view them as indices, but this indices now here represents our logarithms of the corresponding numbers to the basis of 4, so meaning that in case we take this one. Here we can say this one here, 16 here: 16 is the logarithm we can say. 16 is the number to which four must be raised to find two.
This is in index notation, so meaning that we have logarithm of 16 to the base to the base of four we shall find is two so location. 2 is of the number 16 to the base of four and logarithmic theme of let’s say: 60 no 64, but let’s say 256 logarithm. We can write in full battery writing for short form, so locality of 256 to the base to the best of four is four.
As we can see here, we have 256. So the logarithm of this 256 to the base of 4 we shall find is 4 so mean that the logarithm of 256 through this base, you will find that one we can short form or short write that one in a form of log. Now this of 16 will be 16, but if we are dealing with the base of four, so it shall be the best of 4 we shall find? The answer is equal to 2, and here we have the logarithm of 256! The same is to the base of 4, and that one will give us the value as 4. !
Remember here is what we had what we have written here. We refer to it as index notation, but if you write it in this form, here is what we refer to as logarithmic is referred to as logarithmic notation. So if you have numbers written in this form or your computational equation being written in this form, then you are writing it in logarithmic notation! But if it is in this form, it is index notation and the way of changing a number written in index notation to logarithmic notation is just simple in case. We want to write this one in logarithmic, notation you just let say the logarithm of 1 to the base of 4 is equal to the power, so meaning that it will be the logarithm of the number to the base of which you the number is to be raised, and then it will be equal to the power or the index, as we have done here. So it’s just simple in case you want to rewrite this one to index notation. You just say it is the best year which is for to start then, after that you will have to raise it. That number to which you will find that logarithm. So it is 4, then, after finding that one, then your body will be equal to 2 56. That is how we rewrite it back to index notation. Let’s see some of the examples. Let’s check, for example, 3 raised to 4 equals to 81 and we are being informed or asked to write that one into logarithmic notation.
If you want to write that one into logarithmic notation!
Remember what i told you you start with this, so you will say logarithm of 81 to the base of three? So three will be written down. There will be equal to the power to which that base is raised, so the power is 4, so shall be to the power of 4, and in case you are being issued with a number like logarithm of 125 to the base of 5 equals to 3, and you are being asked to find or to rewrite that number into index notation. What you do here is you write 5 down here. This is in logarithmic, notation and the other one which i want to write here? This one here is index notation, so we need to rewrite the one we have here in index notation and writing it in index notation. You start with the base here which is 5! Then you raise it to the number which you are find, as the logarithm of what you have so it will be raised to 3 will be equal to the value here. So you come back to this, so it will be equal to 125. That is how we compute or rewrite a number being index form or index notation to logarithmic notation and vice versa.
That is how we do in case. You have that one. I’ve shown you and this one i’ve shown you now, let’s consider the following numbers. For example, i have 15, i have number like 150. I have another one like 1500, 1 billion, 500, 000 and so on, and also i have a number like 0. 0015!
I have zero point: zero, zero, zero, zero, zero one, five seven and i want to rewrite this number into standard form! You will find out that for the first one, this one here is the same as i write it in a form of 1? 5 multiplied by 10 since the decimal point. I have moved it to this point in order to write it uh into standard form, then my power will be positive, so i will have to power one, and that is the index form of that uh, the standard form of that, but the 10 here, i’ve written it in index form, as you can see, so, meaning that if i take 1.
5 times 10 raised to power 1, i will still get the same value as initial, which was 15. !
The same applies to the second one, and here i will find is 1? 5 times 10, but here the decimal point will move from the extreme end of the right hand, side to the left two places, so it shall be raised to power two, because i’m moving this way to this point. So it shall be that point and we shall have 1. 5 times terrorist power 2 and for this case here, as you can see, you can help me is 1?
5 times 10 raised to power, yeah good, it’s power 3! , so you will find, is to power three and the other one here. What do you think? should i race it to? i will find, is one point five times ten: what power should i raise yeah good? you will find this to power 6, because the decimal point will move from the extreme end of the right hand, side to the left, 6 places, so it shall move from here to here, and that is six places uh. I shall raise it to six, and now this one here, since the decimal point is here, we can move it to a number where we have a number or a whole number so on moving. We are going to move the decimal point to the right and moving it to the right. We shall count the number of places we are moving for this case here we shall have 1! 5 times 10, but since the decimal point has moved to the right our value here, 10, we shall raise it to a negative index such that it shall be to the negative number of places we have moved, and that is to negative three, and for this case here we are going to have it being moved to 1. 57 times? Now you can view it yourself and guess what index you like right, 10, yeah, it’s negative 7. . That is what we have there, because the decimal point will move to the right, 7 places and we shall have 1?
57 times. 10 raised negative 7. , so that is what we have there! What i have written here is what we refer to standard form of the respective numbers. As you can see, this 15 has been written as 1. 5 times 10 raised to power 1. ! This is the standard form of writing 15. !
So in general that means that if you have a number- and you want to write it in a standard form, it will be written in a form of a times. 10 raised to power n, whereby a is a number between a here is a number between one or one itself? It may be a one or a number beyond one, so we say is a number which is one or more than one, but that number is less than ten. So that means it can be one up to nine, but is less than 10 and n. Here is an integer n is an integer, so meaning n is a number which that number should be rest. So that is how we can write a number in a standard form! So i shall give you an example. I have computed here one so you i want you to work out in case you are being given 0.
46 over 10 write that number into standard day!
Four!
What do you find? next? we are going to see how we can find or see powers of ten and common logarithms. Let’s now see the powers of 10 and common logarithms, and here we are going to consider different kinds of numbers of 10 of tens or multiples of 10? For example, we shall consider the numbers such as 1 10 itself. We shall also consider 100, 1000 and so on, and if we write these numbers in index form, we shall find out that the first one here in order to get 1, we shall write this at 10 raised to 1 10 raised to 0 will give us 1, and if you want to get 10, it should be 10 raised to power 1, you will get 10? The same applies to 100, so if it is, 100 shall be 10 raised to power 2, you will get 100? If it is power 3, you will get a thousand, and if it is power, four is ten thousand, and if it is power, five is equal to hundred thousand and this process continues on once that’s how it continues, and we shall find out that in this case here we have zero one. Two, three, four: five, this zero one, two three four five.
We said that the indexes or indices or powers to which 10 are being raised to find the respective or corresponding numbers which are 10, 1 or 100. As we can view out, remember we say this that 0 1 2 3 0, 1, 2, 3 4. We have even 5 and so on? If we could have continued, these are the logarithms. They are the logarithms of 1, 10 hundred a thousand and so on to the basis of 10, the algorithm of the recordings of number one ten hundred a thousand respectively so on to the basis of ten? But sometimes we can find that we have some numbers which don’t lie between one which lies in one ten hundred one thousand and so on. It lies between the two numbers. It may lie between 1 and 10. It may lie between 10 and 100, so it means determining the logarithms of that number or the powers to which 10 must be raised. To find the numbers is a little bit a challenge. So in that case we need a way in which can help us to identify the logarithms of the respective numbers to the best of 10? , with that. That is what why we use the four figure elementary mathematical table, which will help us to evaluate the logarithms of the given numbers? For instance, we shall consider a number such as 70! . In case we have 70. We know 70 here lies between 10 and 100. In between 10 and 100! We have 70, but we know the logarithm of 10 is 1 to the base of 10! , the logarithm of the logarithm of 100. The base of 10 is 2. , so meaning that this one here is the same as writing. 10 raised to 1, and this one is 10 squared, so you will get 100.
So meaning the logarithm of 10 is one algorithm of hundred is two so meaning the logarithm of 70 lies in between 1 and 2. . So we need to compute that value in which we can find the real value order array logarithm of 70. , and in order to find that that the reason why we incorporate what we have already learned, what we refer to as standard deform, and in that case we shall rewrite 70 in standard deform and it shall be 7. 0 times 10 raised to power 1. In this case? If we have this, we shall now say, since we have 10 raised to power 1? This power here is the log to which we shall find that. So we are going to find the log of this, and if we find the logarithm of, we are going to find the logarithm of seven point zero times ten raised to power one, and if that is the case, we shall end up.
Having one now point, this one arises from the logarithm of 10 raised to 1 is 1 because it is 2 power 1. . Now we need to find the logarithm to which, with the base of 10 7 must be raised? 7? 0 must be raised in order to find this number, so we shall find out using the mathematical table and in order to find that we shall view out using the four figure elementary mathematical level you go to, the logarithms of numbers best tell you’ll find it is written. We have logarithm of numbers to best numbers base 10 and it’s written in bracket log 10 x. That way, and then we have a table there. Under this table we have at extreme left hand side, we have x on top and then in between there we have 0 2 9 and then on the other side you will start by having add, there is add, on the other side, a column of add the first arrow of add and then below the row of add! We have one two, nine one, two up to nine: that’s how it has been written? So if you want to find the logarithm of seventy first, you rewrite it in standard form and we have found the first one we will have one then, since 7 is between 1 0 and 10 is between 1 and 10! Then it means what we shall find is less than 0 to 1, since we need to find the logarithm of 7- and we know 7 is a number between 1 and 10, meaning its logarithm should be between 0 and 1.
So we need that value and to get that value we need to use or have an help from the logarithm table.
So that is what we are going to see and then we are going to find the logarithm of 7 to base 10 using the mathematical table and there we are going to view 7.
0. So here you have some of the look.
You have a table here of the logarithm and on top of it heading is logarithm of numbers base 10, as you can view here and in bracket. We have logarithm to base 10 of x, where x represent the number to which you have to find the logarithm?
But here we have to find the logarithm of 7! 0 and on this page, as you can view under the the row, the column of x, you can find 7. 0.
So you have to turn the page up and go to the next page and you will find it still have the same heading?
But in bracket is continued and then locate 7!
0? And since our 7! 0 has no additional digit, then we shall say it is 7.
00 and we have to go to the second section of the table which runs from zero to nine and there we select zero under the column of zero. We run down until we find where it coincides with the row of 7. 0, where you find the row of 7.
0, you take that value and that will be the logarithm of 7. 0 and for this case we find is 0! 845 so meaning to what we have here! We shall add, plus 0.
8451, and if we add that one we are going to have one point, eight four, five one! That is how we do find the logarithm of given numbers which are in between the multiples of ten! So with that, that is the logarithm to which 10 must be rest in order to find 70. ! So in other words, this just simply means that if you take 10 raised to five 1. 8451, you will get your value as 70. . That is what it simply implies? So let’s continue on and find how we can find others.
Here we have a continuation of the previous session, which was powers of 10 and common logarithms.
Now this one is logarithmic, based and under here we are going to consider some numbers, such as 35! 6, and if you have certain number like that, we identify that that number will lie in between 10 and 100, meaning that the logarithm of that number will lie between one and two, that is the logarithm of ten, a logarithm of a hundred two basis of ten now to identify what is the specific logarithm 35. 6! The first thing you need to do here is to rewrite that number into standard form and rewriting in standard form. We shall have 3. 56 times 10 raised to power 1. Now we need to find the logarithm of this number to base 10, and if that is the case, it means we shall find the logarithm of 3? 56 times 10 raised power 1 to base 10, but remember here the reason why we are writing! This number into a standard form is to find this decimal part of the logarithm? So the decimal part of the logarithm shall be found by the whole number being written in the small point here and then what we have here will give us the whole number of the logarithm part so that whole number we certified to characteristic part. So, let’s see how we shall find, since we have that remember, we said in logarithm. If you have, this is like adding. If you have multiplication you will add, according to the law of indesign the first law and here in logarithm, the same will apply such that, if you have now, we need to find the value of here, and in this case we shall have. First, we need to find the logarithm of 10. Then, if we have a multiplication, it shall be addition and then the decimal part of the logarithm here shall be found using the logarithm table, and that is what we shall add to what we shall have found here and here the logarithm of 10 rest one is one and the logarithm of 3. 56 using the mathematical table.
You will have to go to the locality table as initially i have shown you how to find locate 3?
5 under the column of x, after locating 3! 5, go on the second section of the table where it runs from 0 to 9, locate, 6 and you will find 6 after getting 6 use that column of 6 to run down until where it will coincide with the trip, the row of 3. 5, and you will find your value. There is 55 14 and that 55 14 before it at three point: five at three point: five initial day at zero. We have a point so meaning that value is zero point five, five one four. So the value which we have here will be zero point. So on this one, we are going to add 0. 55 for one now we shall add, and we shall find the the value of, or the logarithm of 35. 6 will be equal to z, 1! 5541. That is what we found when we use 3?
5 there of 3? 5 and the column of 6? So that’s what we find if we work out logarithm of 35? 6 to the base of 10 with the help of mathematical table. The same applies to any kind of localism that lies between the multiples of 10.
? You first change it the number into a standard form. Then you use the mathematical table to help you find the decimal part of the logarithm here.
As we can see, this is the logarithm of this number. The whole part which is behind the logarithm behind the decimal point here, is what we refer to characteristic part of the logarithm, and the decimal part here is what we shall refer to as mantissa, it’s known as mantissa, and this is the current characteristic patch, so the whole part or the whole number part of the logarithm here is the characteristic part, and the decimal part is what we refer to as mantissa. We shall be working out later with the sum of the examples, as we continue on next, we are going to see how we can find logarithm of positive numbers less than one, but before that i will give you an exercise to work out, and that is, you will find the logarithm of 792?
4.
In case you have been given a decimal number or a number which is less than 1, meaning that it is having a decimal.
Then you are being asked to find the logarithm of that number, which is a positive number to the base of 10. This is this is how you should work it out. We shall consider an example on that and, for example, i shall take 0!
0003251 in case. We have that number and you have been asked to find the logarithm of that number to base of ten what you are required to do? The first thing is change the number to standard form, and how do you do that remember standard form? we said it takes the kind of a multiplied by 10 raised to n, and if that is the case, then changing that number to that form? We shall rewrite it in form of three point: two: five one times ten raised to the power of which the decimal points are moving to the right and that will be equal to 4.
So, since you have moved to the right, then it means what you have here will be 10 raised to negative 4. So that is what we have and in case you want to find the logarithm of that number to the base of 10! Then this is what happens! First, find the logarithm of 10 raised to negative 4 you will find is the indices. What you will find uh? that index is negative 4! So you will have negative 4 as the characteristic part. Then you have to find the decimal part or what we refer to the mantissa part and the mantissa part. Here you will find it by the help of the table, since the logarithm of 3? 251 lies between 0 and 1 and finding the logarithm of 3? 251! You need a mathematical table to help you find it, and in that case you shall use the logarithm of number base, 10 table and under the column of x, locate 3? 2 after locating 3. 2 move to this! In second session of the table, which runs from 0 to 9, locate, 5 use that column and the row of 3. 2 and find where the two coincides and you will find is 0. 5119! Then we have an additional one move to the third section of the table where it runs from one to nine locate one, and you will find it coincides with three point two at one and the column there. It is add so, whatever you have there, this one is equal to four decimal points or the four decimal places so meaning the first one we found was zero point. Five one one, nine- and here we have one that one is written in this form, whatever it is. In the add column, it is to four decimal places, so on addition, we are going to have 0. 5120! That is what we shall find as the logarithm of three point, two five one. So on this mantissa part, we are going to add this one here, which is positive and on rd we shall have 0. 5120.
So this is the mantissa part, and this is the characteristic part now another point to note: when you are dealing with logarithm which have negative characteristics, it means that only the characteristic part is one which is negative, not the mantissa, and in that case the characteristic part is written with a negative sign on top of it. For example, in computing this we shall write the negative on top of 4 to mean that only the characteristic part is the one which is negative, but point will remain the same, and then we shall have one two zero.
This means that the logarithm of this number is this, but only the characteristic part is negative and the mantissa part is positive, and this number is read as ba four point: five one, two zero?
That is how we do find the logarithm of a number which is positive but less than 1 to the base of 10! .
Next, let’s see how we can find much more now you work out.
I want you using the same technique. Work out, 0. 034 work out the logarithm of that one to the base 10 and see what you will find the logarithm of 0! 034 to best 10? Now under lugar the anti-logarithms.
This is the reverse of what we have doing, what we have done. We shall go the reverse of it, and that simply means that in case you know the logarithm of a given number, then you can go back and find the real number to which the logarithm was found in order to find what you have, and in doing that, we shall have the help of antiloganity table here to help us find the numbers back. So we shall consider a number such as 0?
7125, and that is the logarithm to base 10. A certain number was found to base 10 to give us the value of 0. 7125, meaning that the log to base 10 of x is equal to 0. 7125. What is the number to which that logarithm was found in order to find this number 0. 7125? in order to find that, remember, we say say this: is the x to find that we refer to his anti-locality and writing. It is written in this form that we shall have to rewrite it in index form and writing in index form we should have 10 raised to 0? 7125 is equal to that number which we need. This one here is our anti-logarithm, and if we work out, we shall find the number to which that the number to which it was found to the base of 10 to find 0! 7125. Let’s use the antilog name table and see how it will help us now. We have anti-locating table here and you will find on top of that logarithm table. It has been written at logarithms and in brackets, 10 raised to x, 10 raised to x. Under there we have the same same almost similar, but not same similar to what we had in logarithm table, whereby we have column of x and then another column which runs from zero to nine and then the third one having add, and it runs from one to nine. So what you do the first column of x. It is point it starts with the point and is point zero, zero, zero one. That way so meaning is point zero point: zero, zero. Until then, once so in case of zero point, seven one, two: five! In order to find the ant logarithm, you shall view zero or locate 0. 71, locate, 0. 71 and you will find it on the next page after locating 0. 71 locate!
Two and two is in the second session, a section of the table and then using that second uh, that column of two run down until where you will coincide with the row of 0! 71 and you will find, is 5. 152 5? 152, and then we have a remainder which is 5, so got the third part of the table where we have add locate five using five column!
Our defeat corner run down until where you will find or coincide with uh there of 0! 71, and you will find the value there is six so that six is in three decimal places. So you would add it to this, and you will find you will have five point, one five, eight! So this is the anti logarithm of zero point. Seven one, two five! in other words, we do say this: this is the number to which the logarithm to the base of 10 was found in order to get 0. 7125. That’s how we do evaluate delay and the logarithm the anti-logarithms of given localities.
So we are going to find the anti logarithm of the following logarithm! That was the first example! Second, one is where the the characteristic part is negative and then strip one two, five, the characteristic part, is the one which is negative so to find the ant logarithm here. What you require is first to write it the way we need to identify it much more clearly! So in that case we shall have 10 rest negative 4, since that is the characteristic part, and if you take that one you multiply by 10 raised to 0?
3125.
That means i’m separating the characteristic part from the mantissa part, and that is how we do work it out! So we shall find that way? So here this one already, you know how you will find the value, and here we need the anti-logarithm table to help us evaluate which number can we find in case we find the anti-logarithm of 0! 3 1 2 5, and if we find that when we are going to find using the same procedure, you will find under the column or the row of 3! 1 at 0!
31 column of 2 you will find is 2? 051 2?
051.
Then you will have to add whatever you have, there is 5 and under 5 is 2, so it’s 0. 002 and if you add that students you will find is 2. 053. So that is what we find at this part or the mantissa part is giving us the ant.
Logarithm of that is 2. 053, so we shall have 2. 5 2! 053 multiplied by 10, raised to negative four and remember this. One is the same as rewriting! Since this one is the negative index! You will have two point: zero, five, three multiplied by one over ten thousand, and if that is the case, you know what happens, you will have a value of 0. 0002053. This is the number to which the logarithm was found, the base of 10 to give us by 4.
3125. The same applies to the positive numbers whereby, if it could be positive now, this negative sign could be neglected if it is positive and the same approach applies and you will find it out. So i will give you one example of the positive and i read that one you will work it as out as an exercise that is find the ant logarithm of 2? 4835! Now, let’s see how we can find applications of logarithm or where we can apply logarithms, and then we shall do multiplication and division from the in-laws of indices there. We can really recall what we had learned in laws of indices, and we did say that if you have, for example, a raised to m so we shall relate it to base 10. It was a raise to m multiplied by a raised to n.
We said it is the same as taking a raise to m plus n, but this one we are going to relate it to the base 10 and say that it should be 10 raised to m multiplied by 10 raised to n, and this one will give us 10 raised to m plus n, as according to the first law of indices? That is the sum of two powers and remember the bases are the same, so meaning that if this is the case, the same case applies to the division, and that means we shall have 10 raised to m divided by 10 raised to n, and this should give us 10 m minus n.
That is what we refer to as the difference of two powers as the second law of indices, and this one corresponds to the law of am raised to m divided by a raised to n, and we say it should be a into m minus n! Now we are going to see an application of this uh application of this into multiplication and division! For example, we have a certain uh certain example here, 357 times 47? 9 357 times 47. 9. Now, if we do apply the law of logarithm here and bearing in mind the law of indices, so we shall have this kind, and in that case we are going to have this. One will give us the first one which we have as 357! It will give us 3. 5 7 times, 10 raised to the power 2, and if we write this one in logarithmic form to base 10, let’s write it in index form to base 10 whereby we are going to find the logarithm of this number?
We shall have two points now, the logarithm of 3. 57?
With the help of the mathematical table, you will find the logarithm being five five, two seven. So if you add to these two, you will find you are going to have five five, two seven, so 357 can be still rewritten being rewritten by writing 10 to power of 2? 5527, and now we have 47? 9 this one.
We can still rewrite it as 4. 79 times 10 raised to power 1, and this one in index form! We can still write it as now.
This one will be one point, and now we are going to find the logarithm of that part and if we find the logarithm of 4.
79, with the help of elementary four figure elementary mathematical table, you will find it is six eight zero, three six, eight zero three? So that is what we are going to have. So in the case of having these numbers here is the same as taking this number multiplied by the other number? So since we have the same base now we have created the same base, so we can have 10 raised to five 2!
552 seven multiplied by ten raised to one point: six: eight zero three is equal to now, since it is the same base and it’s a multiplication sign and then this one will be equal to 2. 5527, plus 1! 6803, and if we work out what you get, you are going to find your value as this one here will give us uh. Now you will have 4! 2330. That is what you find as the value of, if you multiply this directly using the law of indecision, and that is the first law. So this is the answer there, but if you go back, what do you find? you will find that this one here is the logarithm of the value which you will find. If you multiply this one directly and if that is the logarithm, we can still find the anti logarithm of this logarithm and find that number to which you will find the logarithm to the base of 10 and find four point, two three three zero, and if that is the case, this is how we do write the ant logarithm. So we are finding the ant logarithm of four point 4. 2330 and that one is the same as writing!
10 raised to 4 multiplied by 10 raised to 0.
23 3 0, and this one will give us this one here. First is 10 raised to 4, and now we are going to multiply by now the anti-logarithm of 0. 2330. You will find it as let’s use the logarithmic table to find it out, and if you find it out, you will have it as one point: seven one: that’s what we are going to find it out and if you multiply here, you are going to have 171 17 100. That is what you will find. That is where you use the mathematical table or the logarithm to compute the value of 357 times 47. 9. So if you compute this mathematic just directly by long division, you are going to have a value which is equivalent to 17 100 point, so you will find that this point three here is minimal, so the we can neglect that one, since what we have is seventeen thousand 100, which is much more close to this, and this one is just with a minimal range of xcd! So we can neglect and use this way of computing it. So that is how we can incorporate the application of logarithm in computing mathematics. So we can continue on with the computation and see how we can value find out a much more of it.
Let’s deal with now division, we have 864 divided by 136 again according to the law of indices.
That is the second law. We can still make this one to the base of 10, but, firstly, we shall work out individually by taking like this 864.
We can rewrite it into 8. 64 times 10 raised to power 2. And if that’s the case, this number can still be written in form of 10, raised to the power 2 point now the logarithm of 8.
64. You find the logarithm of that number so that if you find the anti-logarithm of the number which you will have found, you will still get the same value as 864! . So we are going to find the logarithm of 8.
64 and the logarithm of 8. 64 is 9 3, 6, 5, 9, 3, 6 5, and then again we are going to find the standard form of 136 and the standard form will be 1. 36 times 10 raised to bar 2, and if you work it out, you will find that the logarithm now will be 2 point now.
There’s the ant organism, if you write 10 and then the logarithm above. That is the angle logarithm, but if it’s just the number alone is the logarithm of that one to the base of 10?
. So this is the anti-logarithm. With the strength of 1. 36, now the logarithm of 1.
36. To give us the mantissa part, you will find that the logarithm will be 1, which is 0. 13 point one three, three five, so meaning here. If we have we add here, we shall have two point, one three, three five. So that is what we find remember. We have made the same base, so we can use the law of hindus and this is to work it out and if we apply that law that says that, if we have 10 raised to m divided by 10 raised to n, we shall find 10 raised to m minus n! If that is the case, if we apply, we are going to have this one multiplied by this one or the first, a computation multiplied by the second divided by the second computation, meaning 10 raised to 2. 9365, divided by 10, raised to 12. One three. Three five is equal to ten two point: nine, three, six, five minus two point, one three, three: five, and if you compute that we are going to have, we shall have 10 raised to 0. 803. That’s what we find. So if you find the anti locality of that number, you are going to find six point three five three, and that is the value of 864, divided by 100 136. That is how we do work out when you incorporate logarithm into multiplication and division from the laws of indices. So we shall consider an example which just directly work out there, and before that we are going to consider even different kinds of them. I shall be doing dealing with some of the computation you will find and may encounter, or you may find them difficult in working out such as you may find 3. 14 in case you have a number like 3. 142 being squared and you want to compute it using logarithm.
What you do like this number here. It can be written as 3. 14 multiplied by 3. 14, so that number, if you compute it in form of logarithm, then it should be. You will have the number, which is three point, one four one, four now it’s square, that is the number and remember in working out logarithm. Please emphasize much more on your spacing and working out plan it to be much more needful such that it will not confuse you while working it out. So, for example, here you start with the number, then you write the standard form and then, after the standard form here you put the logarithm of that num for this one we have now in standard form? The number which is in bracket here will be three point one four times ten raised to power: zero, but remember this one here is being squared, so it’s just that way. So you have three point one four times ten raised to power: zero because we are not multiplied by any kind of or 20 such that this 10 raised to power? 0 will give us 1, and if you multiply this, you still get the one in bracket? Then you find the logarithm of that one. After finding the logarithm, you will find out that the logarithm will be equal to. If you use the mathematical table, you are going to find the logarithmic 0. 4 0. 4969, so you will find is zero point four, since here we have zero, so we shall start with that as my piece and our characteristic.
So it’s zero point, four, nine, six, nine, but remember here we have two so 2 means square. We are going to square this and since we have found the logarithm and is 2 base of 10, meaning we have to find this logarithm to base of 10 and then to base of 10 whatever we shall find, because it is to base 10 and is the logarithm?
The sign here will be our addition, since, if we multiply the numbers, the logarithm we do add them together! So in adding together is like, after finding 6 square meaning we should multiply itself twice so the logarithm we find we just add another, one of which is the same as taking this one multiply by two, and if you do that, you are going to find here, you are going to have a value such as 0. 9 0. 993, three, eight. If you go on multiplying and if you do that, that is what you will find as the logarithm of 3. 14 squared and if that is the logarithm there, and then you can go back and find the anti-logarithm of this number so that you will find the number which is here so after finding this. You can still find a number x such that, if you multiply by 10- or let’s just say this is 10 raised to 0, but that number x is in this form, 10 raised to zero point: nine. Nine three eight, this number, if you do it out the logarithm of this number, you will find this. So we need that number? So we are going to use anti logarithm to find the number and you will find.
The number is nine point. Eight! Five. Eight nine point: eight five, eight! so it’s about ten ways to zero. So just that is what you will have.
So if you find the logarithm of this number, you are going to find this, but the anti-local name of that you will find this and if you compute it out directly by a long division, you will find you are having the same! As my point was this in case you have a number being? Squared is just the same as finding the logarithm of that number. After finding the logarithm, then you multiply by the power to which it is being squared! If it is 2 multiply by 2, if it is 3 multiply by 3, if it is the fourth, you have to raise to power 4 after finding the logarithm multiply it by 4, then you find the ant logarithm.
You will find the same value, so that is how we compute, or we apply logarithm in computing of the squares, cubes or even defaults, the fourth route? So, for the rest to force the powers of four five and so on, and now we are going to see how we can find out or work it in a mathematical mixed computation, for example, we’re going to have 6 34 multiplied by 4 36 by 688, divided by 784, and see what we shall find now you have been told to compute this using logarithm, and if that is the case, what you are required to do here is now what i told you about planning planning your work, rearranging it much more well and needful, so that you avoid confusion, and that include you write the number here then the standard form of the number. Then you find the logarithm of that number and the computation of that column! So that is how we do separate the numbers in that category!
So such that like for the first number here, is 6 34. I write my 6 34 here or 634 in standard form?
It will be 6. 34 times 10 raised to power 2. . If i find the logarithm of that number.
I will find this 2 point now. The logarithm of six point three four. I will find is eight zero, zero point, eight zero, two one, then i will continue with this one. Fourth, at six, so right here, four type, six and that is 4. 36 multiplied by 10 raised to power 2 and the logarithm of that number is 2 point. Now then, at the locality of 4. 36 you will find the logarithm of 4. 36 is 0. 6395. So since we have multiplication here, if we find the logarithm we are going to add. So i will add here and if i add i’m going to get here- 6 yeah, 11, 4 14, so after getting 14 now will be 1.
8. So if is that the case, then i will put a point and the one i have to carry? I will take it to the characteristic part- and i add here one so it will be 3 plus two i’ll get five. I will continue with my calculation or down there, so i’m going to multiply by again six eight eight. Now this six eight eight in standard form with six point eight eight times ten raised to power two and this one.
I will now have to write it down here.
So it will be two point starting with that, because that is the logarithm of ten square and then the logarithm of 6! 8 is equal to 6. 88. You will find 6. 88 is 0. 8376 now, after getting that, you go step by step. Since we have a multiplication here, it means the logarithm i found here, i’m going to add today one i will find after computing the value on 688! So i’m going to add, if i add here, i’m going to get 12 years, this is eight, so i’ll find it at nine, and then here will give me seven here, twelve, so the one i have carried, i will add it to the characteristic part there and i end up having eight point two seven nine two, my computation has has not entered so what i’m remaining with, because these are not ended, i’m having the division and the division is a whole division of it. So what i have i have 784 after computing. I have 784. So instead of the form i’m going to have seven point, eight four times ten raised to power two and this one will give you me two point.
Seven point: eight four, seven point: eight seven point: eight 4 is 0. 8943 and since we have a division line here, meaning a quotient rule, then under the laws of indices, we said: if it’s a question, i wrote the best ten. Since we are working to the best of ten we mean we shall subtract the powers, and this is the same as the power of the power to which 10 must be raised to find 784. And if that’s the case, we are going to subtract and on subtraction we should have. We shall borrow one. So you will have 12 minus 3?
You are going to get 17 subtract 9, you will have 8, and here we have 11 subtract now here you can see after subtracting. Here you borrowed 1. Here you find this 17, you subtract. 9, you find eight here you are main with one this one. You can subtract eight from it! So what do you do? you go to characteristic part, and then you borrow one after borrowing, one you bring it with between eleven. After getting eleven, you subtract, eight.
You will remain with three, but remember you borrowed one. So if you put one it remains seven, so seven minus two: you will end up having five getting that that’s the value, what you find after working out this computation in logarithmic form. So after that you can go back and find the value to which will give you that figure you have. So what you do! You will come back and write! This is times 10 raised to power 5, the characteristic part. I will write it here and then the mantissa part it will be raised to the first 10. So that is 10, and this is 0? 3849 quarterly? The characteristic part you will raise it to the 10 of the last and, as we have been doing, then the decimal part or the mantissa part is which you will raise to the first 10, and then you find the ant log of this value and the antilog you use the antelope table.
That is point three! Eight point, three eight point, three, eight four and then nine you’ll find is 2.
421? So i can write 2.
421!
Then you have to add 5. So that is 0. 005, and if that is the case, you are going to have 6. So this is what we are going to find here is the ant node, so i am going to have 2. 426 times now, 10 raised to 5, that is 100 000 and if the 100 000, if you multiply by that, you are going to have your value as you will have your answer as 242 000 600. That is what you find if you compute this numerical problem here using logarithmic form until you find the ant log of the answer- and you will arrive at this- that is how we work it out?
May it be in bar form, for example, the bar one, if, if we are told for example, you may encounter in such a way that after working it out the multiplication there, you find that this answer here is less than what you have to subtract or the division part! For example, let’s say this one was to be 2! 8943!
I’m saying this is an example, and here you had, for example, eight point!
For example, let’s say one seven, two two: if that was the case, then you can view that the characteristic part there is the denominator has having much more than the numerator?
So what you do you just subtract, the mantissa as normal, so the mantissa will give you, for example. Here will be one, and here you will get one! Yes, two, nine minus seven. You will have two eight minus one is seven, but now for the characteristic part, since after multiplication, you have found this one and it’s much more lesser. What you have to subtract here is a subtraction side. So what you do you will just take two, but you subtract, eight! If you take to subtract eight, whatever you will find, is negative, six negative six, but remember that negative only indicate that the characteristic part is the only one which is negative! So what you do you put there a bar side on top of six to indicate that only the characteristic part is the one which is negative and on computing, the antelope?
Then here you should raise to negative 6, but here we remain now the positive part which is 0? 7221. That is how we compute in case you find the one or the quotient is more than the one which you have to subtract. So that’s how you compute, then.
Sometimes there are situations which are rise and you will find, for example, you have to add 2.
49, for example, by 2! 49 plus in your computation, and you find your id to 3 by 3. 53 in case you encountered such mathematics? What you do, you will just add it directly, which is by 2.
49 plus now that one will be here is by 3? 53.
But what i suggest is you separate the characteristic part from the mantissa part such that you will now have bar 2, then plus 0. 49, plus bar three plus zero point, five: three, then you add together and on additional. You will find here. You will have about five plus one point now this one is gonna. Give you zero two. So if you have something of that sort, as you can see now here the characteristic pattern after getting this is positive? So on additional to this, you will now end up having bar four point: zero, two?
That is how we compute those who have in there but or where the characteristic is negative. You should not confuse between the characteristic and mantissa of logarithm after working it out now, let’s learn more about roots and others. We shall be finding either square root, cube root or any other given or stated roots of certain. Given numerical figures, for example, you may be given a number like you have a mathematical problem like 22 multiplied by 56, divided by 63, but here now we have to find the fourth root of this, given sum here for that even condition here what you are required to do, you shall involve indices and logarithm in working out and whatever you shall find out, you have to find the fourth root of that, given numerical figure, for example, working out this, given sum here, if you have been given, may be much like a numerator may come because that even having four kinds of given decimals, maybe you may have a sign of multiplication division, another one, maybe even subtraction or a given. Addition here, whatever you should do, just follow the body mass then from there work out the numerator first, then the denominator, but considering the local themes law together with the indices law. So after then, then you find the fourth root of given that value you have found. We shall consider this example and work out and see what we shall find! For example, if you have been given this kind of given mathematical problem, what you are required to do here, if you involve indices and logarithm, then what you are involved to do you have to construct a table whereby you will have a number. You will have the standard form of that given number, and you will have the logarithm of that number such that you are going to have a table of that kind and remember we have said in case you have this mathematical problem.
You have to deal with the numerator first, then the denominator for this case we shall start with number 22 for which we have here. We write it first in standard default and remember in writing in standard form! We shall be involving this kind such that we shall have 2. 2 multiplied by 10, raised to power 1. Since 22. If you write in style form, you will get that one. I remember for this case here this one here, which we have 10 raised to power. 1 will form the characteristic part of our logarithm, whereby we should have 1. Then this remaining part here? If we have to find the logarithm of that part, we have to find it and it will give us the mantissa part of our number there.
So we shall use the log table to find the the mantissa part uh, where we shall find it from two point two and we shall use the logarithm of numbers base! Ten to find the number two point: two. I will find two point two and a column of zero. You will find it’s zero point, three, four, two four. So we shall add three, two, three, four, two four? So that is what we have there and then we shall go for another number, which is 56. The same applies. We shall have 5. 6 times, 10 raised to power, and here we shall have one point.
Then the current, the mantissa part will be the logarithm to base 10 of 5? 6, and we shall find for 5. 67 for 0. 7482, then remember! We said we have to work with the numerator first and for this case the numerator! We are going to take 22 multiplied by 56, meaning that its logarithm shall be. Since it is multiplication side, then we shall have an addition side in the evaluation of its logarithm. For that case, we shall have here six. Yes, ten here we are going to have nine is ten and that one we shall add characteristic parts such that we shall have this point as 3! 0906?
Then we are going to work out the denominator part, and that is 63. So we shall have 6. 3 times 10 raised to power 1. And in logarithmic form.
It is the one point and then we find the characteristic, but then mantissa part of 6! 3, that is the logarithm of 6. 3, will give us the mantissa part, and that will be 0.
7993. So after having this since we’re having that as denominator- and here we have division sign- then it means for each logarithm. We are going to take a subtraction from the numerator, so it will be six minus three! You remain three here you’re going to have one, yes, nine, and since here we’re having now seven, we need to subtract nine? To do that, you have to borrow one from the characteristic part you put here.
B10, you subtract one you put here or you borrow one and then you bring it to b17. Now that’s 17 you’re going to subtract nine! so if you take it is eighteen, eighteen, minus nine you’re going to get nine such that here mean nine nine subtract! Seven you’ll get your value as two and here two minus one. You will have one so meaning that whatever you’re going to have there you’re going to have 1. 2913. So we are going to find this one uh in this kind, and you will have this one- will be 10 raised to 0. 299 multiplied by 10, raised to the power 1. Why? because the characteristic part will be simply independent and that will be to base 10 and you will find steroids power and the mantissa part is the one which we need to find the ant log so as to find the number which will give us that given value but remember this value here which we were working for, we cannot work this way because we need to find the fourth root of that given number for that case, then it means whatever we are having here. We need to find first uh 1 3 multiplied by a quarter, but if it were that we want, we were not supposed to find this fourth root. Then we could just find directly by use of what we have here, but now we need to find the fourth root and finding the fourth rule we are going to multiply by a quarter because that is like! We are finding the rational in this of that, given value so in finding the fourth root, we are going to multiply by a quarter and remember if this characteristic part is positive, then we shall just multiply directly. But if we, this characteristic part was negative, we could make this number in such a way that it is divisible by four. Let’s work with a case where the characteristic part is positive, and if that is the case, we shall just go directly and we shall find this. One will give us 0. 4 into 20 and into 30. We shall find 4 into 37 28, then five, but this one into just four decimal places. We shall have zero point, four, two, two or six? So that is what we are going to have. So we need to find the ant log of that given number, and if we find the antelope of that number, we shall now be finding the value to which resembles that given number so the antelope we shall find it as for 0. 242? Another column of 2 is 2?
642 2. 65 2, but we have to add there of the column of 6 and that of the column of 6 is 0. 004, so we are going to find the numbers 2. 6.
So this is the value to which you will find if you find the fourth root of the given value here, remember what we have done, work out, the numerator first then the denominator. Then you multiply by the root of the fraction of the root which you are finding?
That means you divide by the given number or integer, which will be the root sign there. Then, from there, you find the ant log to find the given number now.
What would have happened in case after working the numerator and denominator, and you find your value here- is negative!
The characteristic part is negative. If we were told the characteristic part was negative in order to work out, for example, if you have 2 0 1 3, then it means you have to multiply by a quarter, and this means that you will have to make this characteristic part to be divisible by four before dividing it by four and in such a way. In order to find that you will take this characteristic part. You add three say that is the only least possible number to which we can add to one to make it divisible by four and remember, if you add three, this three is regarded to be in same condition with by one since by 1 means that only this characteristic part is negative. Then it implies that even this 3 we are adding can also be negative, because we are adding two characteristic parts such that we shall have by four, and it will not be negative one plus three such that you get to know it should be in this kind and remember whatever your attitude characteristic part should also be applied to mantissa part in such a way that 3? 2013 now this one here is what we are going to multiply by a quarter, and in that case now you shall now get this. One will be by four plus three point: two zero one: three multiplied by a quarter. If you work out you’re going to get this one here, we’ll give you now bar one now plus 4 into 3 is 0, so it will be 0 4 into 32. You’ll get is 8 and then 4 into 0 is 0 again and 4 into 1 is zero, so it shall remain that way.
Point to 13 is into 12, you get three and so on. So that is what you are going to have. So, if you add this, you are going to get by one point: eight: zero, zero. Three now this value here is what you will now required to find the ant log of this, and if you find the analog, is the value which will be corresponding to the respective route, you will be asked or required to find.
So that is how we do work out in the two perspective, one if the characteristic part is positive and two, if the characteristic part is negative, it applies to any given value of root. For example, if it is n, then we could do it with n here and remember, even if it is by one we have to make it in such a way that it is divisible by n. With that we have come to the end of our topic today. I believe it has been interesting, see you in our next videos.
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