Change of Base Formula
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.[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.Answer: 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{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}[/latex]
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.Answer: [latex]\frac{\mathrm{ln}8}{\mathrm{ln}0.5}[/latex]
[ohm_question]86013[/ohm_question]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.Answer: 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-display]\begin{array}{l}{\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{array}[/latex-display]
Try It
Evaluate [latex]{\mathrm{log}}_{5}\left(100\right)[/latex] using the change-of-base formula.Answer: [latex]\frac{\mathrm{ln}100}{\mathrm{ln}5}\approx \frac{4.6051}{1.6094}=2.861[/latex]
[ohm_question]35015[/ohm_question]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.
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- Change of Base Graphically Interactive. Authored by: Lumen Learning. Located at: https://www.desmos.com/calculator/umnz24xgl1. License: Public Domain: No Known Copyright.
CC licensed content, Shared previously
- Question ID 35015. Authored by: Smart,Jim. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
