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:

\[\begin{align} \log_{32}{243} &= \frac{\log_{2}{243}}{\log_{2}{32}} \\ &= \frac{\log_{2}{243}}{5} \\ &= \frac{1}{5}\log_{2}{243}. \\ \end{align}\]

Now applying the exponent property of logarithms:

\[\begin{align} \log_{32}{243} &= \log_{2}{243^\frac{1}{5}} \\ &= \log_{2}{3}.\ _\square \end{align}\]

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, \(\log_a b = \frac{\log_c b}{\log_c a}.\ _\square\)