Solving exponential equations using exponent properties (advanced) | High School Math | Khan Academy

So let's get even more practice solving some exponential equations. I have two different exponential equations here, and like always, pause the video and see if you can solve for x in both of them.

All right, let's tackle this one in purple first. You might first notice that on both sides of the equation, I have different bases. So it would be nice to have a common base. When you look at it, you're like, well, 32 is not a power of 8—or at least it's not an integer power of 8—but they are both powers of 2.

32 is the same thing as 2 to the fifth power, and 8 is the same thing as 2 to the third power. So I can rewrite our original equation as, instead of writing 32, I could write it as 2 to the 5th, and then that's going to be raised to the x over 3 power. That is equal to, instead of writing 8, I could write 2 to the third power, and I'm raising that to the x minus 12.

Now, if I raise something to a power, then raise that to a power, I could just multiply these exponents. So I could rewrite the left-hand side as 2 to the 5x over 3. I just multiply these exponents, and that's going to be equal to 2 to the (3x minus 36).

Now things have simplified nicely. I have 2 to this power is equal to 2 to that power, so these two exponents must be equal to each other. 5x over 3 must be equal to 3x minus 36. So let's set them equal to each other and solve for x.

So, 5x over 3 is equal to 3x minus 36. Let's see, we could multiply everything by three. Let's do that. If we multiply everything times three here, we're going to get 5x is equal to 9x minus 108. Now, we can subtract 9x from both sides, and so we will get 5x minus 9x, which is going to be negative 4x, is equal to negative 108. We're in the home stretch here!

Divide—whoops, sorry about that! We could divide both sides by negative 4. We are left with x is equal to—what is this going to be? 27. x is equal to 27, and we are all done! We're all done, and if you substituted x back in there, you would get 32 to the (27 divided by 3). So, 32 to the ninth power is the same thing as 8 to the (27 minus 12) power, which is 8 to the 15th.

27 minus 12 is 15. So anyway, that was fun. Let's do the next one now. This one looks interesting in other ways. We have rational expressions. We expect we have an exponential up here and an exponential down here.

The key realization here is—well, the first thing I'd like to do—let me put this 25 in terms of 5. We know that 25 is the same thing as 5 squared. So we can rewrite this as 5 to the (4x + 3) over—instead of 25, I could rewrite that as 5 squared, and then I'm going to raise that to the (9 minus x). That, of course, is going to be equal to 5 to the (2x + 5).

Now, 5 to the second and then that to the (9 minus x), I can just multiply these exponents. So this is going to be 5 to the (4x + 3) over 5 to the (18 minus 2x), and that is going to be equal to 5 to the (2x + 5).

Now, let's see, there's multiple ways that we could tackle it. We could multiply both sides of this equation by 5 to the (18 minus 2x)—that's one way to do it—or we could say, "Hey, look! I have 5 to some exponent divided by 5 to some other exponent." So I could just subtract this blue exponent from this yellow one.

The left-hand side will simplify to 5 to the (4x + 3 minus 18 + 2x), and that, of course, is going to be equal to what we've had on the right-hand side: 5 to the (2x + 5).

Now we just have to simplify a little bit. Let's see, this is going to be—in fact, we could just say—look, I'm having trouble with my little pen tool. Whoops! All right, so now we can say this exponent needs to be equal to that exponent because we have the same base.

So what we have here on the left-hand side that I can rewrite as (4x + 3 minus 18 plus 2x)—I'm just multiplying the negative times both of these terms—so plus 2x is going to be equal to (2x + 5).

So there's a bunch of different things we could do here. One, we could subtract 2x from both sides. That'll clean it up a little bit. We could also subtract 5 from both sides, so let's just do that. Let me just subtract 5 from both sides.

I'm skipping some steps here, but I figure you're at this point reasonably comfortable with linear equations. So then, on the left-hand side, we are going to have 4x, and then you have (3 minus 18 minus 5).

3 minus 18 is negative 15, and negative 15 minus 5 is negative 20, which is going to be equal to zero. Because those cancel out, you can add 20 to both sides and get 4x is equal to 20. Divide both sides by 4, and we get x is equal to 5, and we are all done!