What Are Logarithms?

Having explained why we use logarithms, in this post I’ll talk about exactly what logarithms are and how they work.

You’ve probably heard that “logarithms are the opposite of powers”. That’s true but not particularly helpful. Let’s do it by example. If I say:

What is log10 of 1000?

I really mean

10 to the power of what gives 1000?

The answer, of course, is 3:

103 = 1000
=>
log10(1000) = 3

In general, if 10x = y then x = log10(y)

You don’t have to use 10. You can replace it with any number and that number is called the base.

For example I might use a base of 2. So if I say:

What is log2 of 32?

I really mean

2 to the power of what gives 32?

The answer here is 5:

25 = 32
=>
log2(32) = 5

Note that you can never have the logarithm of a negative number.

Now try these questions
(Opens in a new window)

Neither the base nor the the number have to be integers – and nor does the answer. But that doesn’t matter, the principles are the same. If you have a ‘log’ button on your calculator then it probably uses a base of 10. On a calculator the ‘log’ button is shorthand for log10 (‘log base 10’). We often call the base ‘b’, so in general:

bx = y
=>
logb(y) = x

Going Backwards

So how do we get back from logs to ordinary numbers? We use powers – remember, logs are the opposite (inverse) of power. Taking a log then doing a power gets us back to the original number. For example:

log10(1000) = 3
10log10(1000) = 103 = 1000

Or we can do it the other way round: if we do the power first then the log, we get back to our original number again:

103 = 1000
log10(103) = log10(1000) = 3

That’s one of the rules of logarithms I’ll talk about next time in the how to… post.