## The logarithm

The logarithm of a positive number \(x\) is the value that gives you the answer to the following question:

**To what exponent do I need to raise the base \(b\) to get \(x\)?**

So the logarithm (with base \(b\)) of \(x\) is the exponent \(c\) that solves
\[
b^c = x.
\]
If \(c\) solves this equation, we write \(c = \log_b x\).

### Calculation rules

Remember the calculation rules for powers? Because of those, the logarithm has some nice properties. First of all, the logarithm of a product is the sum of the logarithms
\[
\log_b x + \log_b y = \log_b (xy).
\]
The logarithm of a power is the exponent times the logarithm of the base
\[
p \log_b x = \log_b x^p.
\]
Finally, if you want to calculate the logarithm with base \(a\) and you know the logarithm to base \(b\), you can just divide by the \(b\)-logarithm of \(a\):
\[
\log_a x = \frac{\log_b x}{\log_b a}.
\]