# Understanding natural logarithms

## A base 10 logarithm is a power of 10

A base 10 logarithm is a power of 10. If the power is less than 1 then the logarithms are negative.

It also makes sense to have powers of 10 that are fractions, rather than whole numbers. The power doesn’t need to be an actual number; it can be an algebraic expression.

You should begin to see the power of taking logarithms. It’s much easier to solve for ‘normal’ numbers rather than powers.

## Using a calculator to find logarithms to the base 10

We can use a calculator to find the log to base 10 of a number.

If we know the log and we want to know what number is 10 to the power of it then we normally use the ‘second function’ key.

## Multiplying numbers, add the logs

When we multiply two powers of 10, the easiest thing is to add the powers. So if you want to find the log of two-numbers-multiplied-together then you add the logs of each one.

This really shows its use when we use algebra, rather than numbers because you end up with an unknown that is no longer a power.

## Dividing numbers, subtract the logs

To find the log of one-number-divided-by-another, you subtract the log of the denominator from the log of the numerator. Again, this means that your unknown is no longer a power.

## A natural logarithm is a log to the base e

Exponential decay has this natural number e in it. Logarithms to base e are called ‘natural logarithms’. The benefit is that we stop expressions like λt being powers, which makes it easier to solve problems.

It’s important that you understand how to take natural logs of the decay equation and that you see how it’s useful. You’re often given N_{0}, so you can calculate its natural log using a calculator.

## Using a calculator to find natural logarithms

Finding the natural logarithm of a number and finding which number has a particular natural logarithm is very similar to doing the equivalent to base 10.