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 \(a\): \[ \log_a x = \frac{\log_b x}{\log_b a}. \]