Solving Simple Logarithmic Equations - Problem 4

Solving a simple logarithmic equation. So for this particular example we are looking at log base 5 of 5. We don’t have any another term on the other side, but that’s okay because we know that this has to be equal to something. We can call this equal to x if it helps us see our answer.

Whenever we're dealing with a logarithmic equation, put it into exponential form, so we end up with in this case 5 is equal to 5 to the x. So there is really a little hidden exponent here of 1, so whenever we have our bases the same, our exponents have to be the same and this leaves us with x equals 1.

For this actually the x is something we introduced to this problem so we don’t really want that in our answer, so we actually we would just want to say that this is equal to 1.

Now what I want to talk about is is 5 special in this case? Does this have to be log base 5 of 5? What if I had log base 8 of 8, we would still put it into exponential form and get pretty much the same exact set up. In this case we would get 8 to the x is equal to 8, you would then get 1. What this comes up with is actually a property of logs.

If we have log base n of n, n being any positive number, this is going to equal 1. So we’ll talk about why it’s supposed to be positive later, but whenever your base and the number inside the log are the same, it's always going to cancel out leaving you with 1.