Logarithm Laws

If you are studying logarithms it is reasonably safe to assume that you’re already reasonably familiar with index rules; those shortcuts that allow for swift calculation of exponents when dealing with equal bases. The log laws are really just a rearrangement of these.

First, multiplying. Since when you multiply terms with the same base you add the indices, the log of a product is equal to the sum of the logs of the factors of that product. This is best demonstrated starting with the index rule and working through an example.

\begin {aligned}a^m \times a^n &= a^{m+n} \\ 8 \times 16 &= 128 \\2^3 \times 2^4 &= 2^7 \\ \log_2{8} + \log_2{16} &= \log_2{128} \\ \log_a{m} + \log_a{n} &= \log_a{mn}\end{aligned}Index / log law for multiplication / addition

Second, division. As the inverse of multiplication this one is now hopefully easy to spot: subtracting the parts. The log of a quotient is equal to the difference between the logs of the dividend and the divisor (numerator and denominator).

\begin {aligned}\frac{a^m}{a^n} &= a^{m-n} \\ \log{\frac{m}{n}}&= \log{m} - \log{n}\end{aligned}Index / log law for division / subtraction

Using this rule for division it follows that we should be able to find the value of a logarithm of 1. As any value divided by itself is 1, choosing a value expressed in index form allows us to apply the previous rule.

\begin {aligned}\frac{a^b}{a^b} &= 1 \\ a^{b-b} &= 1 \\ \log_a{1} &= b-b \\ \log_a{1} &= 0 \end {aligned}The log of 1 in any base is 0

Terms with indices raised to a power are an extension of the multiplication of indices already described. This approach can be applied to the addition of logarithms:

\begin {aligned}(a^m)^n &= a^m \times a^m \times \ldots \times a^m \\ \log_a{m^n} &= \log_a{m} + \log_a{m} + \ldots + \log_a{m} \\ \log_a{m^n} &= n \log_a{m}\end {aligned}Powers in logarithms


Logarithms are a way of expressing powers.

The logarithm \log_b{x} for a base b and a number x is defined to be the inverse function of taking b to the power x, i.e., b^x.¹

The simplest way to explain the concept of logarithms is through an example:

\begin{aligned}2^5&=32 \\ \log_2{32}&=5\end {aligned}

The subscript of the logarithm above denotes the base of the logarithm. This is equivalent to the number which is raised to a power in the preceding equation. The logarithm to the base 2 of 32 is 5, because 2 to the power of 5 is 32.

The equation \log_3{81}=x can be solved by asking ourselves “what power of 3 gives a result of 81?” In this case the answer will be x=4 and can be calculated relatively easily. For most questions involving logarithms you will be resorting to the functions in your calculator or spreadsheet (or log tables, which have become increasingly rare).

x^y = z \iff \log_x{z} = y

¹ Weisstein, Eric W. “Logarithm.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Logarithm.html