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 $10$ or $e$ , 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 $n\ne 1$ and $b\ne 1$ , we show ${\mathrm{log}}_{b}M\text{=}\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}$ .

Let $y={\mathrm{log}}_{b}M$ . Rewriting in exponential form, we have ${b}^{y}=M$ . We can then take the log base $n$ of both sides of the equation. It follows that:

\begin{matrix} {l o g}_{n} (b^{y}) & = {l o g}_{n} M & Apply the one-to-one property . \\ y {l o g}_{n} b & = {l o g}_{n} M & Apply the power rule for logarithms . \\ y & = \frac{{l o g}_{n} M}{{l o g}_{n} b} & Isolate y . \\ {l o g}_{b} M & = \frac{{l o g}_{n} M}{{l o g}_{n} b} & Substitute for y . \end{matrix}

$\begin{array}{c}{\mathrm{log}}_{n}\left({b}^{y}\right)\hfill & ={\mathrm{log}}_{n}M\hfill & \text{Apply the one-to-one property}.\hfill \\ y{\mathrm{log}}_{n}b\hfill & ={\mathrm{log}}_{n}M \hfill & \text{Apply the power rule for logarithms}.\hfill \\ y\hfill & =\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\hfill & \text{Isolate }y.\hfill \\ {\mathrm{log}}_{b}M\hfill & =\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\hfill & \text{Substitute for }y.\hfill \end{array}$

For example, to evaluate ${\mathrm{log}}_{5}36$ using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.

\begin{matrix} {l o g}_{5} 36 & = \frac{l o g (36)}{l o g (5)} & Apply the change of base formula using base 10 . \\ \approx 2.2266 & Use a calculator to evaluate to 4 decimal places . \end{matrix}

$\begin{array}{c}{\mathrm{log}}_{5}36\hfill & =\frac{\mathrm{log}\left(36\right)}{\mathrm{log}\left(5\right)}\hfill & \text{Apply the change of base formula using base 10}\text{.}\hfill \\ \hfill & \approx 2.2266\text{ }\hfill & \text{Use a calculator to evaluate to 4 decimal places}\text{.}\hfill \end{array}$

Let us practice changing the base of a logarithmic expression from

5

$5$ to base e.

Example

Change ${\mathrm{log}}_{5}3$ to a quotient of natural logarithms.

Show Solution

Because we will be expressing ${\mathrm{log}}_{5}3$ 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 $5$ .

\begin{matrix} {l o g}_{b} M & = \frac{l n M}{l n b} \\ {l o g}_{5} 3 & = \frac{l n 3}{l n 5} \end{matrix}

$\begin{array}{c}{\mathrm{log}}_{b}M\hfill & =\frac{\mathrm{ln}M}{\mathrm{ln}b}\hfill \\ {\mathrm{log}}_{5}3\hfill & =\frac{\mathrm{ln}3}{\mathrm{ln}5}\hfill \end{array}$

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 $n\ne 1$ and $b\ne 1$ ,

{l o g}_{b} M = \frac{{l o g}_{n} M}{{l o g}_{n} b}

${\mathrm{log}}_{b}M\text{=}\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}$ .

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.

{l o g}_{b} M = \frac{l n M}{l n b}

${\mathrm{log}}_{b}M=\frac{\mathrm{ln}M}{\mathrm{ln}b}$

and

{l o g}_{b} M = \frac{l o g M}{l o g b}

${\mathrm{log}}_{b}M=\frac{\mathrm{log}M}{\mathrm{log}b}$

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 ${\mathrm{log}}_{2}\left(10\right)$ using the change-of-base formula with a calculator.

Show 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.

$\begin{array}{c}{\mathrm{log}}_{2}10=\frac{\mathrm{ln}10}{\mathrm{ln}2}\hfill & \text{Apply the change of base formula using base }e.\hfill \\ \approx 3.3219\hfill & \text{Use a calculator to evaluate to 4 decimal places}.\hfill \end{array}$

Think About It

Can we change common logarithms to natural logarithms?

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

Show Solution

Yes. Remember that $\mathrm{log}9$ means ${\text{log}}_{\text{10}}\text{9}$ . So, $\mathrm{log}9=\frac{\mathrm{ln}9}{\mathrm{ln}10}$ .

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 $10$ 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 ${\mathrm{log}}_{b}M$ .

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

Module 14: Exponential and Logarithmic Equations

Learning Outcome

Example

The Change-of-Base Formula

Example

Think About It

Summary

Candela Citations