what exactly *is* a logarithm and what does it do?

Question

I mean, I’ve used them in algebra many times but I never really understood what it does. Kinda like in biostatistics how I could do the math, but how it worked was beyond me entirely.

So yeah: like what’s this sorcery and what does it do/why do we use it?

In: Mathematics

Answer 1

It undoes exponentiation. If we consider the function f_a(x) = a^(x) where a is some positive real we have an injective function. We can see its by writing:

a^(x) = a^(y)

a^(x) a^(-y) = 1

a^(x-y) = 1

and we know that a^0 = 1 (except when a = 1, in that case a^x = 1 for every x) so:

x-y = 0

x=y

So f_a(x) is injective and so has an inverse as long as a isn’t 1.

Lets call the inverse f^(-1)_a.

So f^(-1)_a(f_a(x)) = x. Great!

If we say have an equation:

a^x = b

we can solve it with the inverse

x = f^(-1)_a(b)

This function is special and frequently used enough to deserve its own name the logarithm. So log_a(y) is the inverse of a^x. The convenient exponential identities work backwards with them, lets write ab in a tricky way:

ab = exp(log(a)) exp(log(b)) = exp(log(a) + log(b))

now take the log of both sides

log(ab) = log(a) + log(b)

So lets adress what base of log and exp we should use. The answer that it doesn’t matter, there is a convenient pick e, but why and what value it has doesn’t matter right now. So lets select e so

exp(x) = e^x and ln(x) = log_e(x)

The statement is that you can get any base with these but thats true for any number so for now e is just a placeholder for your favourite number.

We can see how:

a^(x) = exp(x ln(a)) = exp(ln(a))^(x) = a^(x) = exp(ln(a^(x))) => ln(a^(x)) = x ln(a)

and with that:

x ln(a) / ln(a) = ln(a^(x)) / ln(a)

x = ln(a^(x)) / ln(a) => ln(f_a(x))/ln(a) = log_a(y)

So we now see how you can use just ln to get any kind of logarithm. And now we can adress the issue with f_1(x) now having an inverse:

log_1(y) = ln(y)/ln(1)

and

1 = e^0 => ln(1) = 0

so

log_1(y) ~ 1/0 and so is undefined.

The complex logarithm is slightly more exciting and you’d basically get there with the exponential similarly to how we started operating with it in the last part. Hopefully seeing how you can work things out from just the concept of what this thing is helped making the little fella less mysterious.