## Logarithms

### Introduction

The logarithm is the power to which a base must be raised to obtain a particular number. That is:

 $y = b^t \mbox{ iff } t = \log_b y$

### Manipulating Logarithms

The following rules/results make it relatively easy to manipulate logarithms:

1. $\log\quad u v = \log u + \log v$
2. $\log u/v = \log u - \log v$
3. $\log u^{k} = k \log u \mbox{ for } u \gt 0$
4. $\log_{b}u = (\log_{b}a) (\log_{a}u) \mbox{ for } u \gt 0$
To prove this result let $u= a^{p}$ (so that $p=\log_{a}u$). Then:
$\log_{b}u = \log_{b}a^{p} = p\log_{b}a = (\log_{b}a) (\log_{a}u)$
5. $\log_{b}u = \frac{1}{\log_{u}b}$
6. Combining the previous two rules yields the following “change of base rule”:
$\log_{b}u = \frac{\log_{a}u}{\log_{a}b} \mbox{ for } u \gt 0$
7. $\frac{d}{du} \log_{b}u= \frac{1}{( u \ln b)}$ where $\ln$ denotes the natural logarithm (i.e., $\log_{e}$).