A review of logarithms

The best way to understand logarithms is through an example. If you take 10 to the third power (10 x 10 x 10) the result is 1000. The logarithm is the inverse of that power function. The logarithm (base 10) of 1000 is the power of 10 that gives the answer 1000. So the logarithm of 1000 is 3.

You can take 10 to a negative power. For example, taking 10 to the -3 power is the same as taking the reciprocal of 103. So 10-3 equals 1/103, or 0.001. The logarithm of 0.001 is the power of 10 that equals 0.001, which is -3.

You can take 10 to a fractional power. Ten to the one-half power equals the square root of 10, which equals 3.163. So the logarithm of 3.163 is 0.5.

Ten to the zero power equals 1, so the logarithm of 1.0 is 0.0.

You can take the logarithm of any positive number. The logarithm of values between zero and one are negative; the logarithms of values greater than one are positive. The logarithms of zero and all negative numbers are undefined; there is no power of 10 that gives a negative number or zero.

The examples above use base 10 logarithms, because the computations take 10 to some power. You can compute logarithms for any power. Mathematicians use natural logarithms, using e (2.7183), but natural logarithms are rarely useful when graphing data.

Biologists sometimes use base 2 logarithms, often without realizing it. The base 2 logarithm is the number of doublings it takes to reach a value. So the log2 of 16 is 4 because if you start with 1 and double it four times (2, 4, 8, and 16) the result is 16. Immunologists often serially dilute antibodies by factors of 2, so often graph data on a log2 scale.