Most calculators can only evaluate common logarithm ([latex]\mathrm{log}[/latex]) and natural logarithm ([latex]\mathrm{ln}[/latex]). In order to evaluate logarithms with a base other than [latex]10[/latex] 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.
change-of-base formula
The change-of-base formula can be used to evaluate a logarithm with any base.
For any positive real numbers [latex]M[/latex], [latex]b[/latex], and [latex]n[/latex], where [latex]n\ne 1[/latex] and [latex]b\ne 1[/latex],
To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms.
Given any positive real numbers [latex]M[/latex], [latex]b[/latex], and [latex]n[/latex], where [latex]n\ne 1[/latex] and [latex]b\ne 1[/latex], we show
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]
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 [latex]n[/latex], 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 [latex]e[/latex].
Rewrite the log as a quotient using the change-of-base formula:
The numerator of the quotient will be a logarithm with base [latex]n[/latex] and argument [latex]M[/latex].
The denominator of the quotient will be a logarithm with base [latex]n[/latex] and argument [latex]b[/latex].
Change [latex]{\mathrm{log}}_{5}3[/latex] to a quotient of natural logarithms.
Because we will be expressing [latex]{\mathrm{log}}_{5}3[/latex] as a quotient of natural logarithms, the new base [latex]n = e[/latex].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 [latex]3[/latex]. The denominator of the quotient will be the natural log with argument [latex]5[/latex].
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.Evaluate [latex]{\mathrm{log}}_{2}\left(10\right)[/latex] using the change-of-base formula with a calculator.
According to the change-of-base formula, we can rewrite the log base [latex]2[/latex] 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 [latex]e[/latex].
[latex]\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]
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.