Logarithms

A logarithm is the inverse of a power. The statement a raised to the power x equals N can be rewritten as: the logarithm of N to the base a is x. So log to base 2 of 8 is 3, because 2 raised to 3 is 8. The base must be positive and not equal to 1, and you can only take the logarithm of a positive number. Reading a logarithm out loud, you are really asking 'what power gives this number?'

Before calculators, multiplying or dividing large numbers by hand was slow and error-prone. Logarithms turn multiplication into addition and division into subtraction, which are far easier. You look up the logs of your numbers, add or subtract, then look up the answer in reverse with an antilog table. This is exactly why log tables were printed in every textbook for over three hundred years.

Three laws do the real work. The product law says the log of a product is the sum of the logs. The quotient law says the log of a quotient is the difference of the logs. The power law says the log of a number raised to a power equals that power times the log of the number. Two special values help everywhere: the log of 1 is always 0, and the log of the base itself is always 1.

A common logarithm uses base 10 and is written simply as 'log'. Every common log splits into two parts. The characteristic is the whole-number part and comes from the position of the decimal point: for a number with n digits before the decimal point, the characteristic is n − 1. The mantissa is the decimal part, always positive, and you read it from the log table. To reverse the process and find the number itself, you use an antilog table. Mastering this by hand is the skill this chapter trains.