### 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]

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*.

- The numerator of the quotient will be a logarithm with base

### Example: Changing Logarithmic Expressions to Expressions Involving Only Natural Logs

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

### 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.