Today's soundtrack is Cryptopsy: None so Vile, an oldschool technical death metal album with vicious vocals from the legendary Lord Worm.
Today, we'll finally learn what these mysterious "logarithms" are. The "LOG" key has been sitting on your calculator, taunting you; when you press it, your calculator does not turn into a log; nay, it remains a calculator, a marvelous piece of technology to be sure, but subtly a liar, for it does not transform into a piece of organic matter. But no more! Fear not, fair brother or sister, dear reader; today, we will learn. Oh yes.
We
will
learn.
So let's start off by experimenting a bit. If we hit the [LOG] key and then type a number, it outputs another number. It's a kind of black box, but just playing around with the calculator doesn't immediately reveal any function or pattern. log(10) = 1, log (200) = 2.301029996, log(0) gives a domain error, log(-1) gives a nonreal answer error, log(0.1) = -1...
What's actually happening here? Let's try to figure it out. Here's a hint - it's something to do with powers.
10¹ = 10; log(10) = 1.
10² = 100; log(100) = 2.
10³ = 1000; log(1000) = 3.
Ah! So here's what our black box is doing. When we use the LOG function, it takes a number, converts it to base 10, and gives us the exponent that 10 must be raised to! To put it most simply, log x = y means x = 10^y, with x > 0 because there is no exponent that will make a positive number (10) negative. Logarithms calculated based on Base 10 are called Common Logarithms.
There are two forms that we can write answers in when asked to determine a logarithm: exponential form and logarithmic form. If asked to determine the logarithm of 100, here's how we could write the answer:
Exponential form
100 = 10^2
Logarithmic form
log 100 = 2
We can also determine logarithms with a base other than 10. We use the form "logax = y," which means "x = a^y." There are a few restrictions that apply here: x > 0, a > 0, and a ≠ 1.
An example of a logarithm with base 3 is log39 = 2, which in exponential form would be 9 = 3^2.
A couple of interesting points: Any time we see a log with a and x of the same values, our answer will be 1. Any time x is 1, our answer will be 0.