 So one important application of exponential equations is to find logs to different bases. And this goes back to the notion of the one true log. The base of our logarithms can be anything, but there are only a few useful bases. If 10 to the power a equals b, then log no base indicated b equals a, and we call these common logarithms. Another base that has some cache among computer scientists is if 2 to the power c is equal to b, then log to base 2 of b is equal to c. These are logs to base 2, and these are occasionally used by computer scientists. But the reality is there is no good reason to use any base other than e. We'll do it anyway. So let's say I want to find log to base 7 of 25. If we want to solve this, we can go back to our definition of log. We'll let log to base 7 of 25 be x, and our definition allows us to rewrite this equation. And so I want to solve 7 to power x equals 25. So we'll hit both sides with the log, apply the power rule, and solve for x. And this leads to the process of base conversion. Suppose you know that the log of b is equal to a. So remember this ln means that we're talking about logs to this base e, this funny irrational number. Given this information, can you find the log of b, that's the log to base 10 of b, or the log to base 2 of b? And the answer is you can't, unless you remember the definition of the logarithm and do a little algebra. So for example, suppose I know that the log of b is equal to 3, find the log of b. So again, this ln means that we're working with logs to base e, this log without a base written indicates that we're working with base 10. So we don't know where to start, so let's start with what we can write and worry about whether it's relevant later. We're given the fact that log b is equal to 3, and since these are the logs to base e, then we can say that e to power 3 is equal to b. Next, since we don't know log b, that is our log to base 10 of b, we should let it be an unknown value x, so we can set up an equation. So we have x equals log b, but this means that b is 10 to power x. And now here's a useful thing, I have two different expressions for b. On the one hand, I have e to the third equals b, and on the other hand, I have b equals 10 to power x. Because I have two different expressions for b, we can write down an equation. We'll start off with the obvious statement b is equal to b, and remember, equals means replaceable. So here, I have the claim that anytime I see b, I can replace it with e to the third. So how about replacing it here? And here, I have the claim that anytime I see b, I can replace it with 10 to power x. So I'll replace it here. And that gives me an equation, e to power 3 is 10 to power x. Well, now let's hit both sides with the log. So over on the left hand side, I have log of e to the third. Over on the right hand side, I have log of 10 to the x. I can use my power rule to simplify. Log to base e is just 1, and I can solve for x. And remember, equals means replaceable. I was looking for x, and so I found it. And of course, math ever generalizes, you should generalize this. If you know that the log of b is equal to the a, then the log to base k of b is equal to... Well, I'll let you figure that part out.