# Music and Math Project

## 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 $$a$$: $\log_a x = \frac{\log_b x}{\log_b a}.$