 So far, we've solved different types of equations by using inverse operations. We have an inverse operation for addition, subtraction, multiplication, and division. But what about other operations? So remember, we can characterize expressions by the last operation performed. An exponential expression is an expression which has a variable expression as an exponent. So 3 to power x is an exponential expression. x to power 3, however, is not an exponential expression because while it has an exponent, the exponent is not a variable. If we have an exponential equation, we need an inverse for an exponential expression. So we'll make the following definition. Let a be greater than 0. If a to power b equals c, then log to base a of c is equal to b. Now, a here is the base, and in general, it can be any positive number. However, there are two important cases. First of all, if that base is 10, we obtain what are called common logs. So if 10 to power b is equal to c, then log c is equal to b. And the thing to notice here is that we're not specifying that base of 10. Common logs aren't actually that important. Far, far, far, far, far, far, far, far, far, far more important are what I know as natural logs. And this is based on a very specific real number. Suppose e to power b equals c, where e is a real number that's about 2.71828 and so on. The reason that this particular number shows up will become apparent in calculus. Then we write log ln c equals b. So remember, equals means replaceable, and in each of these we have b is the same as the log. But in every case, b is the exponent. So the thing to remember is that logarithms are exponents. So let's see if we can find a log. Let's find the log to base 2 of 8. So it helps to approach the problem as follows. We want to find the log to base 2 of 8. Well, let's give it a name. How about x? And we want to solve this equation for x. Definitions are the whole of mathematics. All else is commentary. So if log to base 2 of 8 is equal to x, our definition says that 2 to power x should be 8. Since we've been genetically programmed from birth with an inborn knowledge of logs, well, actually, since we don't have any other choice, we can try guess and check and try out different exponents. So we'll find 2 to the x for different exponents, and we find that 2 to power 3 gives us 8. So x equals 3. So our exponent is 3, and we might summarize our results as follows. Since 2 to power 3 equals 8, then log to base 2 of 8 is equal to 3. Or how about log of 0.01? The only important difference here is that there's no base specified. And so remember, if the log has no base specified, the implied base is 10. So again, it's helpful to set up an equation. Log 0.01 equals, I don't know, call it x. Our definition of log says that 10 to power x is 0.01. And since we're not born with the knowledge of logs, we'll go ahead and evaluate different powers of 10. So 10 to the 0 is, 10 to the 1 is. Now keep in mind, we want to get 0.01. And notice that if we have a positive exponent, our numbers are going to be way too big. So we'll have to try some negative exponents. How about 10 to the minus 1? And 10 to the minus 2 is, which is what we're looking for. And so we can summarize our results. Since 10 to power negative 2 is 0.01, then log 0.01 is negative 2. What's important to recognize is that while we can find some logs this way, most logs turn out to be irrational numbers. So let's see if we can find an approximate value for log base 5 of 100. So again, we'll set up our equation. Log base 5 of 100 is, I don't know. But this allows me to rewrite this equation as, and now we'll use guess and check. I'll find various powers of 5. And since 5 to the second is 25, which is too small, but 5 to the third is 125, which is too big, I might conclude that x is between 2 and 3.