# logarithm

• the logarithm of a positive number: the power of the base that produces the number

• common logarithm: a logarithm whose base is 10 $\mbox{log} N$

• natural logarithm: a logarithm whose base is e $\mbox{ln} N$

## properties of logarithms (obtained by applying the definition)

1. $\mbox{log}_b 1 = 0$

A logarithm of 11 equals 0.

2. $\mbox{log}_b b = 1$

A logarithm of the base equals 1.

3. $\mbox{log}_b b^n = n$

A logarithm of a power of the base equals that power

4. $b ^ {\mbox{log}_b N} = N$

A base, to a logarithm power with the same base, equals the antilogarithm.

## laws of logarithms (used to expand or simplify/condense log experession

1. $\mbox{log}_b MN = \mbox{log}_b M + \mbox{log}_b N$
product/sum law
2. $\mbox{log}_b \frac{M}{N}= \mbox{log}_b {M} - \mbox{log}_b {N}$

quotient/difference law

3. $\mbox{log}_b x^n = n\mbox{log}_b x$

power/coefficient law

4. $\mbox{log}_b M = \mbox{log}_b N$ if and only if $M=N$

equal logs law

## change-of-base formula

$\mbox{log}_b N = \frac{\mbox{log}_a N} {\mbox{log}_a b}$ where a is the new base