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Study Guides > College Algebra

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 ee, 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 n1n\ne 1 \\ and b1b\ne 1\\, we show

logbM=lognMlognb{\mathrm{log}}_{b}M\text{=}\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\\

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

{logn(by)=lognMApply the one-to-one property.ylognb=lognMApply the power rule for logarithms.y=lognMlognbIsolate y.logbM=lognMlognbSubstitute for y.\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}\\

For example, to evaluate log536{\mathrm{log}}_{5}36\\ using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.

{log536=log(36)log(5)Apply the change of base formula using base 10.2.2266 Use a calculator to evaluate to 4 decimal places.\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}\\

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 n1n\ne 1 \\ and b1b\ne 1\\,

logbM=lognMlognb{\mathrm{log}}_{b}M\text{=}\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\\.

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.

logbM=lnMlnb{\mathrm{log}}_{b}M=\frac{\mathrm{ln}M}{\mathrm{ln}b}\\

and

logbM=logMlogb{\mathrm{log}}_{b}M=\frac{\mathrm{log}M}{\mathrm{log}b}\\

How To: Given a logarithm with the form logbM{\mathrm{log}}_{b}M\\, use the change-of-base formula to rewrite it as a quotient of logs with any positive base nn\\, where n1n\ne 1\\.

  1. Determine the new base n, remembering that the common log, log(x)\mathrm{log}\left(x\right)\\, has base 10, and the natural log, ln(x)\mathrm{ln}\left(x\right)\\, 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 log53{\mathrm{log}}_{5}3\\ to a quotient of natural logarithms.

Solution

Because we will be expressing log53{\mathrm{log}}_{5}3\\ as a quotient of natural logarithms, the new base, = 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.

{logbM=lnMlnblog53=ln3ln5\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}\\

Try It 13

Change log0.58{\mathrm{log}}_{0.5}8\\ to a quotient of natural logarithms.

Solution

Q & A

Can we change common logarithms to natural logarithms?

Yes. Remember that log9\mathrm{log}9\\ means log109{\text{log}}_{\text{10}}\text{9}\\. So, log9=ln9ln10\mathrm{log}9=\frac{\mathrm{ln}9}{\mathrm{ln}10}\\.

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

Evaluate log2(10){\mathrm{log}}_{2}\left(10\right)\\ 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.

{log210=ln10ln2Apply the change of base formula using base e.3.3219Use a calculator to evaluate to 4 decimal places.\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}\\

Try It 14

Evaluate log5(100){\mathrm{log}}_{5}\left(100\right)\\ using the change-of-base formula.

Solution

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