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Guides d'étude > College Algebra

Change of Base

Learning Objectives

  • Rewrite logarithms with a different base using the change of base formula

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 n≠1n\ne 1 and b≠1b\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{array}{l}{\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{array}

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{array}{l}{\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{array}

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 n≠1n\ne 1 and b≠1b\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 n≠1n\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: Changing Logarithmic Expressions to Expressions Involving Only Natural Logs

Change log53{\mathrm{log}}_{5}3 to a quotient of natural logarithms.

Answer: Because we will be expressing log53{\mathrm{log}}_{5}3 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.

logbM=lnMlnblog53=ln3ln5\begin{array}{l}{\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{array}

Try It

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

Answer: ln8ln0.5\frac{\mathrm{ln}8}{\mathrm{ln}0.5}

Try it

The first graphing calculators were programmed to only handle logarithms with base 10. One clever way to create the graph of a logarithm with a different base was to change the base of the logarithm using the principles from this section. In the graph below, you will see the graph of f(x)=log⁡10xlog⁡102f(x)=\frac{\log_{10}{x}}{\log_{10}{2}}. Follow these steps to see a clever way to graph a logarithmic function with base other than 10 on a graphing tool that only knows base 10.
  • In the next line of the graph, enter the function g(x)=log⁡2xg(x) = \log_{2}{x}
  • Can you tell the difference between the graph of this function and the graph of f(x)f(x)? Explain what you think is happening.
  • Your challenge is to write two new functions h(x), and k(x)h(x),\text{ and }k(x) that include a slider so you can change the base of the functions. Remember that there are restrictions on what values the base of a logarithm can take. You can click on the endpoints of the slider to change the input values.
https://www.desmos.com/calculator/umnz24xgl1

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