In the section on multiplying and dividing using powers, we concentrated on writing numbers as a multiplier and a power of ten. Alternatively the numbers could be written with fractional powers of 10 as:

325 = 10^{2.512}, 625 400 = 10^{5.796}, 0.00256 = 10^{-2.592}

This is more complicated than power notation and requires a calculator or set of tables and is principally of use when dealing with logarithms:

Logarithms are the reverse of powers.

The logarithm is the power of a number to a particular base. Taking ten as the base, and writing the logarithm to base 10 as 'log_{10}':

log_{10}(100) = log_{10}(10^{2}) = 2

log_{10}(1000) = log_{10}(10^{3}) = 3

log_{10}(0.01) = log_{10}(10^{-2}) = -2

We saw above that an alternative way to write numbers is to use fractional powers of 10, so that
325 = 10^{2.512}, 625 400 = 10^{5.796} and 0.00256 = 10^{-2.592}.

We can therefore express these numbers as logarithms to base 10:

log_{10} (325) = 2.512

log_{10} (625 400) = 5.796

log_{10} (0.00256) = -2.592

Why bother? Well logarithms are useful for a number of reasons, one of which is to do complex multiplications without a calculator. When multiplying two numbers you can arrive at an answer by looking up the log_{10} of the two numbers, adding the logs and then looking up the antilog_{10} of the result. In today's world of calculators this technique is not used much, but again it is sometimes a quick way to estimate the result of awkward and complex calculations. Occasionally, you may use logarithms and antilogarithms when solving equations. For instance:

If 10^{x} = 625 400,

then *x* = log_{10} (625 400) = 5.796

Similarly, you can work out a power of a number using logarithms. To work out the square of 5624, first find the log_{10} (3.75) and then multiply this by 2 (the power). The antilog_{10} of the result (7.5) is 3.162 x 10^{7}, which agrees with the answer that my calculator gives (31 629 376). The reverse of this operation is to find the square root of the answer, which can be written as (3.162 x 10^{7})^{0.5}. Dividing the log_{10} (7.5) by two gives a result of 3.75, whose antilog_{10} equals 5624, which is where we started!

As with powers, logarithms can have different bases. A common base used for both powers and logarithms is the constant *e*. Logarithms using the base *e* are called natural logarithms or Naperian logarithms and use the symbol ln (not log_{e})

Powers and logarithms using base *e* are very useful for calculation of population growth, which is covered in a separate resource on the NuMBerS site. For the mathematically curious, this also contains an explanation of *e*.