What’s A Logarithm? And Why Do We Need Them?

What’s A Logarithm? And Why Do We Need Them?

And Why Do We Need Them? You probably learned about them in your high school math class and then never thought about them again.

You probably learned about them in your high school math class and then never thought about them again. I admit, to this day, logarithms (logs for short) are not my best friend. They’re not that intuitive to think about. But they are really useful, so I’ve learned to grudgingly embrace them.


Definition

A logarithm is the answer to the question what** power x** do I need to apply to the base b in order to obtain the number y:

log_b(y) = x

is another way of specifying the relationship:
b^x = y

Let’s plug in some numbers to make this more clear. We will do base-10, so b=10.

log_10(100) = 2

The base-10 logarithm of 100 is 2 because:
10^2 = 100

It’s basically asking how many b’s do I need to multiply together to get y. To get 100, I just need to multiply two 10s together.

This is actually a really nice way of dealing with multiplicative sequences.

Assume something is growing at a changing rate denoted by rn. Over 3 years, its total growth rate is:

(1+r1)*(1+r2)*(1+r3)
In log-scale, we would just take the log of the entire thing:
(I will use log to denote log_10 to simplify notation)
One rule of logs is that log(A*B*C) = log(A) + log(B) + log(C) so:
log{(1+r1)*(1+r2)*(1+r3)}
= log(1+r1) + log(1+r2) + log(1+r3)
Now let's think about log(1+r1) - it's asking what power do I need to apply to 10 so that it equals (1+r1):
10^z1 = 1+r1, so log(1+r1) = z1
10^z2 = 1+r2, so log(1+r2) = z2
10^z3 = 1+r3, so log(1+r3) = z3
This allows us to rewrite each term, and the previous equation simplifies to:
log(1+r1) + log(1+r2) + log(1+r3)
= z1 + z2 + z3
Thus, in log-space, a multiplicative sequence becomes an additive one:
log{(1+r1)*(1+r2)*(1+r3)} = z1 + z2 + z3

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