Logarithms

Logarithms are another way of writing indices.

log

You may often see ln x and log x written, with no base indicated. It is generally recognised that this is shorthand.

log

Laws of logs
The properties of indices can be used to show that the following rules for logarithms hold:

log

Example:
Simplify: log 2 + 2log 3 - log 6
= log 2 + log 3² - log 6
= log 2 + log 9 - log 6
= log (2 × 9) - log 6
= log 18 - log 6
= log (18/6)
= log 3

NB: In the above example, I have not written what base each of the logarithms is to. This is because for the laws of logarithms, it doesn't matter what the base is, as long as all of the logs are to the same base.

Another important law of logs is as follows. This is a very useful way of changing the base (in this formula, the base does matter!). Most calculators can only work out ln x and lg x (usually just written as 'log' on the button) so this formula can be very useful.

log
log