OK, that’s logs sorted. But what’s all this ‘e’ stuff about? What on earth is log_{e}x meant to mean?

It’s the same as any other log. It’s just that the base isn’t 10 or 2 or 4, it’s a number called ‘e’. Which leads to an obvious question…

## What Is e?

Good question, but at Higher you don’t need to know the answer (it’s all explained in Advanced Higher). For now you only need to know that e is a magic number in the same way that π is a magic number. In the same way that π is about 3.141, e is about 2.718.

And in the same way that π is strongly connected with circles, e is useful for working out the sort of decay relationships like half-life that you get in the real world.

So in the same way that log_{10}5 means “what power do we have to raise 10 to to get 5”, log_{e}5 means “what power do we have to raise 2.718 to to get 5”.

Obviously you can’t do this in your head unless you’ve got some weird mathematical super-power. You have to use a calculator. Most calculators don’t have a “log_{e}” button, probably because the lettering would be too small to read on the button. Instead they use the symbol “ln” which stands for “log natural”.

So the “ln” button on your calculator just means “log_{e}“. (The ordinary “log” button usually means “log_{10}“.)

## Reversing Natural Logs

In order to get rid of a log we raise the base to the power of the value. So the opposite of log_{10} is 10^{x}. In the same way, the opposite of log_{e} is e^{x}.

e^{x} is sometimes written as exp(x).