Change of Base Formula

The change of base formula is used to re-write a logarithm operation as a fraction of logarithms with a new base.

The change of base formula $$\log_a b = \frac{\log_c b}{\log_c a}$$

The most common use of the change of base formula is to compute logarithms on a calculator when the only logarithm operations available are $\log_{10}(\cdot)$ and $\ln(\cdot).$ The change of base formula can also be used to simplify logarithm operations.

Many scientific calculators only have two logarithm operations available: $\log_{10}(\cdot)$ $($often displayed as just $\log)$ and the natural logarithm $($with base $e),$ $\ln(\cdot).$ Thus, you can't compute a logarithm operation directly on this calculator unless it has base $10$ or base $e.$ The change of base formula allows you to re-write any logarithm operation with a base of $10$ or $e,$ allowing you to compute the logarithm on the calculator.

Compute $\log_4(13)$ on a calculator using the $\boxed{\log}$ button.

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Recall that the $\boxed{\log}$ button on most calculators represents $\log_{10}(\cdot).$ Applying the change of base formula:

$$\log_4(13) = \frac{\log_{10}(13)}{\log_{10}(4)}$$

Then, the operations on the right side of the equation can be inputted into a calculator to obtain the result:

$$\log_4(13) \approx 1.850219859.\ _\square$$

The change of base formula can sometimes be used to simplify logarithm operations.

Simplify $\log_{32}{243}.$

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We can see that $32$ is a power of $2.$ Applying the change of base formula:

$$\begin{aligned} \log_{32}{243} &= \frac{\log_{2}{243}}{\log_{2}{32}} \\ &= \frac{\log_{2}{243}}{5} \\ &= \frac{1}{5}\log_{2}{243}. \\ \end{aligned}$$

Now applying the exponent property of logarithms:

$$\begin{aligned} \log_{32}{243} &= \log_{2}{243^\frac{1}{5}} \\ &= \log_{2}{3}.\ _\square \end{aligned}$$

The change of base formula is derived using several other logarithm properties.

Derive the change of base formula: $$\log_a b = \frac{\log_c b}{\log_c a}$$

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Let $a,$ $b,$ and $c$ be positive real numbers.

1. Let $\log_a{b} = x.$

2. Rewrite in exponential form: $b = a^x.$

3. Take the $\log_c$ of both sides of the equation: $\log_c{b} = \log_c{a^x}.$

4. Apply the exponent property of logarithms: $\log_c{b} = x\log_c{a}.$

5. Divide both sides of the equation by $\log_c{a}$: $\frac{\log_c{b}}{\log_c{a}} = x.$

Thus, $\log_a b = \frac{\log_c b}{\log_c a}.\ _\square$