166 Use the change-of-base formula for logarithms

Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than 10 or [latex]e[/latex], we use the change-of-base formula to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs.

To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms.

Given any positive real numbers M, b, and n, where [latex]n\ne 1 \\[/latex] and [latex]b\ne 1\\[/latex], we show

[latex]{\mathrm{log}}_{b}M\text{=}\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\\[/latex]

Let [latex]y={\mathrm{log}}_{b}M\\[/latex]. By taking the log base [latex]n[/latex] of both sides of the equation, we arrive at an exponential form, namely [latex]{b}^{y}=M\\[/latex]. It follows that

[latex]\begin{cases}{\mathrm{log}}_{n}\left({b}^{y}\right)\hfill & ={\mathrm{log}}_{n}M\hfill & \text{Apply the one-to-one property}.\hfill \\ y{\mathrm{log}}_{n}b\hfill & ={\mathrm{log}}_{n}M \hfill & \text{Apply the power rule for logarithms}.\hfill \\ y\hfill & =\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\hfill & \text{Isolate }y.\hfill \\ {\mathrm{log}}_{b}M\hfill & =\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\hfill & \text{Substitute for }y.\hfill \end{cases}\\[/latex]

For example, to evaluate [latex]{\mathrm{log}}_{5}36\\[/latex] using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.

[latex]\begin{cases}{\mathrm{log}}_{5}36\hfill & =\frac{\mathrm{log}\left(36\right)}{\mathrm{log}\left(5\right)}\hfill & \text{Apply the change of base formula using base 10}\text{.}\hfill \\ \hfill & \approx 2.2266\text{ }\hfill & \text{Use a calculator to evaluate to 4 decimal places}\text{.}\hfill \end{cases}\\[/latex]

A General Note: The Change-of-Base Formula

The change-of-base formula can be used to evaluate a logarithm with any base.

For any positive real numbers M, b, and n, where [latex]n\ne 1 \\[/latex] and [latex]b\ne 1\\[/latex],

[latex]{\mathrm{log}}_{b}M\text{=}\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\\[/latex].

It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs.

[latex]{\mathrm{log}}_{b}M=\frac{\mathrm{ln}M}{\mathrm{ln}b}\\[/latex]

and

[latex]{\mathrm{log}}_{b}M=\frac{\mathrm{log}M}{\mathrm{log}b}\\[/latex]

How To: Given a logarithm with the form [latex]{\mathrm{log}}_{b}M\\[/latex], use the change-of-base formula to rewrite it as a quotient of logs with any positive base [latex]n\\[/latex], where [latex]n\ne 1\\[/latex].

  1. Determine the new base n, remembering that the common log, [latex]\mathrm{log}\left(x\right)\\[/latex], has base 10, and the natural log, [latex]\mathrm{ln}\left(x\right)\\[/latex], has base e.
  2. Rewrite the log as a quotient using the change-of-base formula
    • The numerator of the quotient will be a logarithm with base n and argument M.
    • The denominator of the quotient will be a logarithm with base n and argument b.

Example 13: Changing Logarithmic Expressions to Expressions Involving Only Natural Logs

Change [latex]{\mathrm{log}}_{5}3\\[/latex] to a quotient of natural logarithms.

Solution

Because we will be expressing [latex]{\mathrm{log}}_{5}3\\[/latex] as a quotient of natural logarithms, the new base, n = e.

We rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log with argument 3. The denominator of the quotient will be the natural log with argument 5.

[latex]\begin{cases}{\mathrm{log}}_{b}M\hfill & =\frac{\mathrm{ln}M}{\mathrm{ln}b}\hfill \\ {\mathrm{log}}_{5}3\hfill & =\frac{\mathrm{ln}3}{\mathrm{ln}5}\hfill \end{cases}\\[/latex]

Try It 13

Change [latex]{\mathrm{log}}_{0.5}8\\[/latex] to a quotient of natural logarithms.

Solution

Q & A

Can we change common logarithms to natural logarithms?

Yes. Remember that [latex]\mathrm{log}9\\[/latex] means [latex]{\text{log}}_{\text{10}}\text{9}\\[/latex]. So, [latex]\mathrm{log}9=\frac{\mathrm{ln}9}{\mathrm{ln}10}\\[/latex].

Example 14: Using the Change-of-Base Formula with a Calculator

Evaluate [latex]{\mathrm{log}}_{2}\left(10\right)\\[/latex] using the change-of-base formula with a calculator.

Solution

According to the change-of-base formula, we can rewrite the log base 2 as a logarithm of any other base. Since our calculators can evaluate the natural log, we might choose to use the natural logarithm, which is the log base e.

[latex]\begin{cases}{\mathrm{log}}_{2}10=\frac{\mathrm{ln}10}{\mathrm{ln}2}\hfill & \text{Apply the change of base formula using base }e.\hfill \\ \approx 3.3219\hfill & \text{Use a calculator to evaluate to 4 decimal places}.\hfill \end{cases}\\[/latex]

Try It 14

Evaluate [latex]{\mathrm{log}}_{5}\left(100\right)\\[/latex] using the change-of-base formula.

Solution

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