# Logarithms

To understand logarithms, you should first understand exponents because a logarithm is an exponent.

Finding a logarithm (base b) of a number is like answering the question:

To what power do I have to raise b to get this number?

Or, in math notation:

logb a = x means bx = a

Examples:

log7 49 = 2, since 72 = 49

log2 32 = 5, since 25 = 32

log10 0.01 = – 2, since 10-2 = 0.01

This can be a little confusing. Remember that the base in the logarithm equation (the small subscripted number) is also the base in the power equation (the number raised to the power), and they stay on the left side.

In real life, we usually only use two bases: log10, also called the common logarithm, and loge, where e 2.71828, also called the natural logarithm.

Log10x is usually written log x, with the base understood to be 10.  Logex is written ln x.

## Properties of Logarithms

The properties of logarithms are analagous to the properties of exponents.

 logb b = 1 for any b Since b1 = b logb 1 = 0 for any b Since b0 = 1 logb 0 is undefined for all b Since there is no x for which bx = 0 logb x is undefined if x is negative It may seem like log-2 –8 should equal 3, since (–2)3 = –8. But on the other hand, log-2 –4 doesn't mean anything: the equation (–2)x = –4 has no solution. logb xy = logb x + logb y Since bm · bn = bm + n Since logb xy = y · logb x Since (bm)y = bmy