Learning Outcomes
- Rewrite logarithms with a different base using the change of base formula.
Using the Change-of-Base Formula for Logarithms
Most calculators can only evaluate 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]. Converting to exponential form, we obtain [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]
applying the one-to-one property of logarithms and exponents
In the demonstration above, deriving the change-of-base formula from the definition of the logarithm, we applied the one-to-one property of logarithms to
[latex]b^y=M[/latex]
to obtain
[latex]\log_nb^y=\log_nM[/latex].
The application of the property is sometimes referred to as a property of equality with regard to taking the log base [latex]n[/latex] on both sides, where [latex]n[/latex] is any real number.
Recall that the one-to-one property states that [latex]\log_bM=\log_bN \Leftrightarrow M=N[/latex]. We take the double-headed arrow to mean if and only if and use it when the equality can be implied in either direction. Therefore, it is just as appropriate to state that
[latex]M=N \Leftrightarrow \log_bM=\log_bN[/latex], which is what we did in the derivation above. That is,
[latex]b^y=M \Leftrightarrow \log_nb^y=\log_nM[/latex].
The same idea applies to the one-to-one property of exponents. Since [latex]a^m=a^n \Leftrightarrow m=n[/latex], it is also true that given [latex]m=n[/latex], we can write [latex]q^m=q^n[/latex] for [latex]q[/latex], any real number.
This idea leads to important techniques for solving logarithmic and exponential equations. Keep it in mind as you work through the rest of the module.
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 Of 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.
tip for success
Even if your calculator has a logarithm function for bases other than [latex]10[/latex] or [latex]e[/latex], you should become familiar with the change-of-base formula. Being able to manipulate formulas by hand is a useful skill in any quantitative or STEM-related field.
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.
Use an online graphing tool to plot [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.
- 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.
