Logarithms

Logarithms (or logs) is just another way of talking about powers. It works like this: Suppose certain variables are related by $N = a^x$. The variable $x$ is the power or exponent and $a$ is the base. Then the log is defined as

$$\log_a \, N = x.$$

We often use base 10 logs, in which case ``$\log_{10}$'' is just ``$\log$''. Some specific examples:

$$100 = 10^2 \rightarrow \log 100 = 2$$

$$3 = 10^{0.477} \rightarrow \log 3 = 0.477$$

Some useful log rules:

$$\log (x+y) \;{\rm is \; irreducible.}$$

$$\log (xy) = \log x + \log y$$

$$\log (x/y) = \log x - \log y$$

$$\log (x^y) = y\,\log x$$