Logarithm Bases

Logarithm Bases

From the definition of a logarithm we know that the base of a logarithm must be a positive number and it cannot be equal to \(\text{1}\). The value of the base influences the value of the logarithm. For example, \({\log}_{2}2\) is not the same as \({\log}{2}\) and \({\log}_{f}{11}\) is not the same as \({\log}_{g}{11}\), (\(f \ne g\)).

We often calculate the “common logarithm”, which has a base \(\text{10}\) and can be written as \(\log_{10}{x} = \log{x}\). For example, \(\log{8} = \log_{10}{8}\).

The “natural logarithm”, which has a base \(e\) (an irrational number between \(\text{2.71}\) and \(\text{2.72}\)), can be written as \(\log_{e}{x} = \ln{x}\). For example, \(\log_{e}{5} = \ln{5}\).

Special logarithmic values

• \(\log_{a}{1} = 0\) \begin{align*} \text{Given the exponential form } a^{n} &= x \\ \text{we define the logarithmic function } \log_{a}{x} &= n \\ \text{So then for } a^{0} &= 1 \\ \text{we can write } \log_{a}{1} &= 0 \end{align*}
• \(\log_{a}{a} = 1\) \begin{align*} \text{From the general exponential form } a^{n} &= x \\ \text{we define the logarithmic function } \log_{a}{x} &= n \\ \text{Since } a^{1} &= a \\ \text{we can write } \log_{a}{a} &= 1 \end{align*}