 The introduction of logarithms allows us to solve two new types of equations. One is a logarithmic equation, and the other is an exponential equation. Now, before we continue, we can use any positive number as a base for our logarithms, but we like to use this irrational number E, 2.7, 1, 8, 2, 8, and so on. The reason that this is a nice base will become clear when you take calculus. So with logarithms, we can solve a new type of equation. If x is an expression containing a variable, and a, b are real numbers with a greater than 0, then a to power x equals b is an exponential equation. Now, if we hit both sides with the log to any base k, we have log to base k of a to the x equals log to base k of b. But we have a power rule for exponents, so this exponential expression x can come out front, and we can solve for that expression as, and since my assumption a and b are some real numbers, the right-hand side is just a number. Since the base doesn't matter, it's convenient to use ln, our base E. So for example, let's say we want to solve 5 to power x equals 10. Well, we'll start off by hitting both sides with the log. Since I have 5 to power x, I can rewrite that using the power rule, and I can solve for x. It's important to recognize that log 10 and log 5 are real numbers. We might not know what the values are, but they have very definite values. Finally, we can cancel out the logs, but that would be wrong. Remember, you can only cancel if numerator and denominator are products. Log 10 is not a product. This is not log times 10. Likewise, log 5 is not a product, and since numerator and denominator are not products, we can't cancel. We have to leave it in this form. How about this equation? We need this in the form a to power x equals b. So we can rearrange things by dividing by 10. And let's do a little bit of arithmetic work. We can simplify 25 divided by 10 is 2.5. We'll hit both sides with a log. We can apply the power rule. This exponent 0.1t can come out front. We have this expression log of E. So the thing to remember is that since E to power 1 is equal to E, then the log of E is equal to 1. Then I can simplify the left-hand side a little further. And now I can solve for t. And again, because the numerator is not a product, it's not log times 2.5. We can't simplify this any further, and we have to leave it in this form.