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Study Guides > Intermediate Algebra

Read: Change of Base

Learning Objectives

  • Use properties of logarithms to define the change of base formula
  • 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]. 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}{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's 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.

Answer:

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].

[latex]\begin{array}{c}{\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]

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, 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.

Answer:

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 e.

[latex]\begin{array}{c}{\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]

 

Think About It

Can we change common logarithms to natural logarithms? Write your ideas in the textbox below before looking at the solution. [practice-area rows="1"][/practice-area]

Answer: 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].

 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.

Given 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.

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  • Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..