# Use the change-of-base formula for logarithms

Most calculators can evaluate only 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

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

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.

### 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],

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.

and

### How To: Given a logarithm with 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 13: Changing Logarithmic Expressions to Expressions Involving Only Natural Logs

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

### Solution

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.

### Try It 13

Change [latex]{\mathrm{log}}_{0.5}8\\[/latex] to a quotient of natural logarithms.

Solution### 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 14: 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.

### Solution

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*.

### Try It 14

Evaluate [latex]{\mathrm{log}}_{5}\left(100\right)\\[/latex] using the change-of-base formula.

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