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Logarithms

Logarithms are exponents that are relative to a given base. Calculations involving multiplication, division, raising to powers and extraction of roots can usually be carried out more easily with the use of logarithms. Logarithms contain three parts: the number, the base, and the logarithm. In the following logarithm example, the number is 1000, the base is 10 and the logarithm is 3.

There are two types of logarithms that appear most often. The first type has a base of ten like the example. The second type has a base of e where e ~ 2.718. Since these logarithms appear so often, they are abbreviated. For a logarithm with a base of 10, the base is not written and it is assumed. For a logarithm with a base of e, it is abbreviated to ln, also with no written base.

Rules of logarithms

It is possible to change the base of a logarithm. This is helpful when using bases that are not the two most common bases. This makes it possible to change the base of the logarithm so that it can be calculated using a calculator since most calculators only have the two bases.

It is also possible to split a logarithm apart. This becomes more useful when variables are involved. It also becomes useful later when graphing data on a log scale and finding an equation for the line. If you evaluate this rule in terms of the multiplication rule for exponents, it becomes easy to see why this rule is true.

Using the rule of multiplication, logarithms can be evaluated with exponents. If the number contains an exponent, it is possible to pull that outside of the logarithm