Logarithms

I'm baaaack! But this time I'm not talking about Chem: I'm here to talk about maths. About logs, to be exact. Mathematical logs. I'll never know why maths and chemistry like using the names of other objects to name stuff (I'm looking at you, 6.022 x 10^23, that number that chemistry named "the mole"), but at least in this case "log" is an abbreviation of something. Anyway, this is rather off topic.

I'm sure in your mathematical career you've encountered indices or exponentials. You know, those numbers to the power of something. Square numbers. Cubic numbers. Quartics. And so on. Well, logarithms are just the reverse of indices. Just like how multiplication and division are opposite processes.

Now, in exponentials, the big number at the bottom is called the base and I'm pretty sure the little number in the top right is the power. So in 2^3, the 2 is the base and the 3 is the power. Now, you see, logarithms also have different bases, as indicated in a little subscript next to the word "log"...

log_{2 = base 2 log}

log_{3 = base 3 log}

_{and so on. The most common base is base 10, so if you see "log" without a subscript, it's base 10 that's being referred to.}

_{Now, as I said before, logarithms are the opposite of exponentials. In exponentials, you have (base)^(power) = answer, but what happens if you have the base and the answer, but not the power? Well, this is where logs come in.}

Let's start simple. 2^x = 8. Now, log_{2 8 = x. Do you see what I did there? I put the base as a subscript next to the word "log" and then the answer. (I'm sure there's another term for this which is less vague than "answer," but I don't know what it is.) x = 3 because 2^3 = 8.}

_{But wait, you might ask. What happens if you get given some crazy numbers so you can't work out what x is in your head? Well, this is where log laws come in.}

You see, logs are the opposite of indices, so log laws are the opposite of index laws. If you multiply two exponentials together, you add the indices together. Conversely, if you add two logs together, you multiply the number next to the word "log" (not the subscript, I mean, the other number. I really need to know what this number's called...). If you divide an exponential by another, you subtract the indices. Conversely, if you subtract two logs, you divide the indices. There's explanations for these but I don't really want to put them up until I know what that damn other number is called, because I'm probably already confusing you enough as it is.

There's also another important log law to learn. If that-number-which-I-don't-have-a-name-for is an exponential (e.g. 2^3 or 3^5 or whatever), then you can rearrange it like this:

log_{2 (2^3) = 3} log_2 2

_{Basically I've just taken the power down to the front.}

_{Two more things that you need to know: if the base is equal to that number, then the answer's 1. That's because when you raise any base to the power of 1, the result is the same as the base. Also, if that number is 1, then the answer's 0, because anything to the power of 0 is 1.}

_{Okay. That was a really confusing explanation, mainly because I don't have a name for that number, and I really need a name that's less vague than "answer." One day I'll clean up this post. One day...}