Formula. Change of Base (logarithm) - Math

We can recognize logarithms as "fractions of exponential". Multiplying the same number with nominators and denominators, we can rewrite fractions with a different expression. Likewise, we rewrite logarithms with different expressions.

Formula. Change of Base (logarithm)

\[ \log_a{x}=\frac{\log_b{x}}{\log_b{a}} \]

Using that formula, you can change the base of the logarithm and add (or subtract) logarithms with different bases, such as $\log_2 3 + \log_4 5$. So the change of base logarithm formula is the base of calculating logarithms.

Remark

Before proving the formula, we should understand the below formula.

\[ (1) \quad a^{\log_a{x}}=x \] \[ (2) \quad \log_a{x^n}=n\log_a{x} \]

Proof

This proof is a bit technical.

\[ (\log_a{x})(\log_b{a})\\ =\log_b{a^{\log_a{x}}}\ \ \cdots\ (2)\\ =\log_b{x}\ \ \cdots\ (1) \]

Dividing by $\log_b{a}$, we get the formula.

\[ \log_a{x}=\frac{\log_b{x}}{\log_b{a}} \]

Note

Physics often uses the base 10 and data are usually plotted in log scale charts with the base 10.