Change of base formula

The change of base formula is a formula for expressing a logarithm in one base in terms of logarithms in other bases.

For any positive real numbers $d,a,b$ such that neither $d$ nor $b$ are $1$ , we have

$\log_b a = \frac{\log_d a}{\log_d b}.$

This allows us to rewrite a logarithm in base $b$ in terms of logarithms in any base $d$ . This formula can also be written

$\log_b a \cdot \log_d b = \log_d a.$

Use for computations

The change of base formula is useful for simplifying certain computations involving logarithms. For example, we have by the change of base formula that

$\log_{\frac{1}{4}} 32\sqrt{2} = \frac{\log_2 32\sqrt{2}}{\log_2 \frac{1}{4}} = \frac{\frac{11}{2}}{-2} = -\frac{11}{4}.$

Special cases and consequences

Many other logarithm rules can be written in terms of the change of base formula. For example, we have that $\log_b a = \frac{\log_a a}{\log_a b} = \frac{1}{\log_a b}$ . Using the second form of the change of base formula gives $\log_b a^n = \log_b a \cdot \log_a a^n = n \log_b a$ .

One consequence of the change of base formula is that for positive constants $a, b$ , the functions $f(x) = \log_a x$ and $g(x) = \log_b x$ differ by a constant factor, $f(x) = (\log_a b) g(x)$ for all $x > 0$ .

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Change of base formula

Use for computations

Special cases and consequences