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.

$$(1)a^{{\log }_{a}x}=x$$

$$(2){\log }_a{x}^n=n{\log }_{a}x$$

Proof

This proof is a bit technical.

$$\begin{gathered} ({\log }_{a}x)({\log }_{b}a) \\ ={\log }_b{a}^{{\log }_{a}x}\cdots (2) \\ ={\log }_{b}x\cdots (1) \end{gathered}$$

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.