Exponential Equations (Equating Indices)

So in this video i’m going to show you a process that we follow called equating indices, there are other ways to solve them as well, but we’re going to have a look at how we solve them through equating indices so say on this problem here to equate indices. What we actually need is?

We need both sides of our equation here to have the same base. So in this case we’ve got a base of two and we’ve got the number thirty-two.

If i can rewrite thirty-two as an exponent with a base of 2, then i can now equate the two indices, so, for example, so it’s still to the power of x 32 is the equivalent of 2 to the power of 5. Now, because i’ve got two bases are the same? What it means is the two exponents now have to be the same for this to be true. So what i can therefore say here is my x must be equal to five.

These two exponents must be equal to each other when the two bases are the same. So if we apply that logic over here as well, we can also do it when we our number over here, is a fraction because remember they’re, just negative indices. So if i’m looking at this i’m gonna rewrite this with the base of five, so it will be five.

The x plus two is equal to now! 25 is 5 squared now, because it’s on the bottom of the fraction i’m going to make a negative indices. So this is 5 to the negative 2 now, because i’ve rewritten this, with both the same base, the base of five i now know that the exponents must be equal to each other. So therefore, my x, + 2, must also equal, negative 2 and once i’ve got this? I can now follow my normal solving equation rules that i’ve done before to find the value of x. So if i subtract 2 from both sides, i find that my x here for this to be true, is equal to negative 4. Now you don’t always get situations where the base has been given to us already.

Sometimes we actually have to follow that process of rewriting it. With the same base on both sides of the equations such as here so i can’t really write 27 at least nicely with a base of nine easily, but looking at both sides of the equation. I know that nine has a base of three quite nicely and 27 has a base of three quite nicely as well. So if i now look at rewriting this, this will be 3 squared! But of course that still has to multiply my x up here and over on this side, 27 is 3 cubed, so i can now rewrite this to say. Well, this is going to be 3 times 2x, because multiplying the two indices is equal to 3 to the power of 3! So that means that my 2x must equal 3, because the base is in the other same, the exponents must equal with each other?

So if my 2x is equal to 3, dividing both sides by 2, therefore, my x must be equal to 3 over 2. For this to be true now, when you follow this process, it’s very important that we do be very careful with our order of operations and making sure we don’t make mistakes.

So let’s have a look at this example.

Here this example here i can write, rewrite both sides with a base of 2.

So if i’m looking at doing that, this left-hand side becomes 2 to the power of 3, but that’s got to multiply. You can see here that just needs to multiply with these here so that’ll be multiplying, x, minus 2 and on the right here! This is the same as saying it’s to the power of negative 1. So what i can now state here is because i’ve got both sides with a power of 2? This 3 bracket, x minus 2, must be equal to negative 1 and i? Can here i use my inverse operations to find the value of x. So if i’m doing that, i’m just going to multiply the 3 out so there’ll be 3. X minus 6 is equal, negative 1, add 6 to both sides, so 3 x will be add.

6 here will be 5 and then divide by 3 will leave my x to equal 5 over 3. So, as you can see here when we’re solving asians we’re the exponent part of the equation is now a variable.

We can use a process of equated indices to be able to find that value of the variable now what’s important here before we equate indices, we need to represent both sides of the equation with the same base, once we’ve done that we can state that the two indices have to be equal with each other! For that to be true, [music]! !