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 [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{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}[/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{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}[/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].
- 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.
- 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 [latex]{\mathrm{log}}_{5}3[/latex] to a quotient of natural logarithms.
Try It
Change [latex]{\mathrm{log}}_{0.5}8[/latex] to a quotient of natural logarithms.
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: 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.
Try It
Evaluate [latex]{\mathrm{log}}_{5}\left(100\right)[/latex] using the change-of-base formula.
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 [latex]f(x)=\frac{\log_{10}{x}}{\log_{10}{2}}[/latex].
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 [latex]g(x) = \log_{2}{x}[/latex]
- Can you tell the difference between the graph of this function and the graph of [latex]f(x)[/latex]? Explain what you think is happening.
- Your challenge is to write two new functions [latex]h(x),\text{ and }k(x)[/latex] 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