# Logs = Exponent-busters

In a word: when you have a problem with exponents, try logs. The great thing about logs is that they take everything down a notch. Multiplication gets busted down to addition. Exponents get busted down to multiplication.

If you're already comfortable with manipulating log expressions, just skip to the next page. Otherwise, here is a quick reminder and attempt to rationalize that exponent-busting magic.

## What is a log?

What logs tell us is the magnitude of a number -- how much space the number takes up on paper.

If the log is 3, then the original number is a '1' followed by three zeros.

If the log is 4, then the original number is a '1' followed by four zeros.

If the log is between 3 and 4, then the original number is something bigger than '1' followed by three more digits.

## How do logs turn multiplication into addition?

So far so good. Let's take it another step. When we multiply 2 normal numbers, we're basically ADDING their magnitudes. Therefore

100 * 1000 = 100,000

The number 100 has 2 zeros, 1000 has 3 zeros, and their product has 2+3=5 zeros. By not-coincidence, the MAGNITUDE of 100 is 2, the magnitude of 1000 is 3, and the magnitude of 100,000 is 5.

**The moral of the story so far: logs allow us to deal with numbers as magnitude, so it turns multiplication into addition.**

## How do logs turn exponents into multiplication?

As you know, using an exponent is a short-hand way of indicating repeated multiplication.

2^{5} = 2 * 2 * 2 * 2 * 2

So logs turn (repeated) multiplication into (repeated) addition

log(2^{5}) =

log(2 * 2 * 2 * 2 * 2) =

log(2) + log(2) + log(2) + log(2) + log(2)

Another way of saying "repeated addition" is "multiplication"

log(2^{5}) = 5* log(2)

**So the whole moral of the story is: logs knock exponents down to multiplication, and they knock multiplication down to addition.**

Bottom line: logs are a biologist's best friend...!

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