 Okay so let's look at an introduction to logarithms and so we have this fundamental definition of what a logarithm is and this goes back to any time we can write something as a power. So if I have a to power n equals b what I have immediately is I could rewrite this expression as log to base a of b is equal to n and there's our basic definition. Now a couple of terms that are worth noting look at the base of the exponent so here a is my base, a is the thing that's being raised to power n. It's the same as the base of logarithms so this expression with base a gives me an expression log to base a. Likewise the exponent, the actual exponent that I'm using here is the same thing as the log and finally the power, the thing that is the exponential, the power is the argument. Now if you haven't heard the term argument used in this sense before what it means is the argument is the thing that we're taking the log of. So this is log to base a of b and there's the argument the argument is b and b is the same as the power. Again be careful with the loose definition the power is this which is also equal to b. The exponent is n, the power is the whole thing. So there's two special bases that are worth considering. If the base is 10, so again if the base is 10 is the base of the exponent or the log is 10 then I don't need to write down the base itself I can just write down log L, O, G, and and what this means log n, no base specified, we read this to mean the same thing as log to base 10 of N and these are known as common logarithms. Why are they known as common logarithms? Well once upon a time they were actually pretty common but much much much much much more important is if the base is E, and E is a real number that's approximately 2.718 mumble, and the meaning of this actually shows up in calculus. It's the basis E. We change our expression entirely instead of writing out L of G, we write LN, and when I write LNN, log of N, this means the same thing as the log to base E of N. Sometimes people will call this the natural log of N, but really it is the one and only log that's actually important. So the way to read LNN, this is not natural log of N, this is not ln of N, as some people might read it, this is just simply log of N, and everything else we should specify the base on. Log of N, log base 10 of N, log to base A of B. So let's see what we can do with these exponential logarithmic expressions. Paper is cheap so we should write things down. We'll write down our definition of logarithms and compare to what we have, what we have with our definition. So our definition says A to power N equals B gives us log to same base, argument is the power, the actual log is the exponent. So this is A to power N equals B gives the log to base A of B is equal to N. So our first expression, 2 to power 5 equals 32, our base is 2, so that's the base of the logs. The argument, the thing that the power is, the power is equal to 32, so B is 32, and our exponent N is going to be 5, and so that allows me to write this as a logarithmic expression, log to base A of B equals N, log to base 2 of 32 equals 5. Likewise, for my second expression, my base is the base of the logs, that's going to be E, my power is the argument, is 10, my exponent is the log, is the exponent, and so log to base E of 10 is equal to x, and again, since we're working base E, I could write ln 10 equals x, so there's my expression. Well, let's see if we can evaluate a few logs. So say I want to evaluate log to base 5 of 125. Now, just to make sure all of our p's and q's are dotted, or something like that, what we might want to do here is I might want to say, well, I don't know what this is, but I can at least give it a name, log to base 5 of 125 equals x, and I will try to solve for x, and how I can do that? Well, paper is cheap, so I'll write down the definition of logarithms and see if I can rewrite the equation. So again, my definition of logarithms, A to power N equals B gives me log to base A of B equals N, and if I compare my definition with what I have, then that tells me that my base A, base of the logs, is 5, my power argument 125, and my exponent, the actual value of the log, is equal to x. So 5 to power x equals 125. Now, at this point, there's not really much we can do other than to try out different possibilities for x. And so here it helps to know some of the powers of 5, either if we know them offhand, or because they are just products of multiples, they're products of 5s, we can calculate them directly. 5 to power 1 is 5, 5 to power 2, that's 2, 5s multiplied together, 5 times 5 is 25, 5 to power 3, that's 5 times 5 times 5 is 125, and oh, there it is, 5 to the power 3, 125, 5 to power x, 125, x must be equal to 3. And so x equals 3, and put things back where you found them, log 5, 125 equals x, x is 3, any place I see x, I can replace it with 3, so I'll replace it back here.