Change of Base

Learning Outcome

  • Change the base of logarithmic expressions into base 10 or base e

Most calculators can evaluate only common and natural logs. 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.

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]. Rewriting in exponential form, we have [latex]{b}^{y}=M[/latex]. We can then take the log base [latex]n[/latex] of both sides of the equation. It follows that:

[latex]\begin{array}{c}{\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}{c}{\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]
Let us practice changing the base of a logarithmic expression from [latex]5[/latex] to base e.

Example

Change [latex]{\mathrm{log}}_{5}3[/latex] to a quotient of natural logarithms.

We can generalize the change of base formula in the following way:

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]
As we stated earlier, the main reason for changing the base of a logarithm is to be able to evaluate it with a calculator. In the following example, we will use the change of base formula on a logarithmic expression, and then evaluate the result with a calculator.

Example

Evaluate [latex]{\mathrm{log}}_{2}\left(10\right)[/latex] using the change-of-base formula with a calculator.

Think About It

Can we change common logarithms to natural logarithms?

Write your ideas in the textbox below before looking at the solution.

In the following video, we show more examples using the change-of-base formula to evaluate logarithms.

Summary

For practical purposes found in many different sciences or finance applications, you may want to evaluate a logarithm with a calculator. The change of base formula will allow you to change the base of any logarithm to either [latex]10[/latex] or e so you can evaluate it with a calculator. Here we have summarized the steps for using the change of base formula to evaluate a logarithm with the form [latex]{\mathrm{log}}_{b}M[/latex].

  1. Determine the new base n, remembering that the common log, [latex]\mathrm{log}\left(x\right)[/latex], has base [latex]10[/latex], and the natural log, [latex]\mathrm{ln}\left(x\right)[/latex], 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.