Change of Base

Learning Objectives

Rewrite logarithms with a different base using the change of base formula

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 $e$ , 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 $y={\mathrm{log}}_{b}M$ . By taking the log base $n$ of both sides of the equation, we arrive at an exponential form, namely ${b}^{y}=M$ . It follows that

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

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.

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

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

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

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.

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

and

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

How To: Given a logarithm with the form ${\mathrm{log}}_{b}M$ , use the change-of-base formula to rewrite it as a quotient of logs with any positive base $n$ , where $n\ne 1$ .

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.

Example: Changing Logarithmic Expressions to Expressions Involving Only Natural Logs

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

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.

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

Try It

Change ${\mathrm{log}}_{0.5}8$ to a quotient of natural logarithms.

Solution

\frac{l n 8}{l n 0.5}

$\frac{\mathrm{ln}8}{\mathrm{ln}0.5}$

Q & A

Can we change common logarithms to natural logarithms?

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

Example: Using the Change-of-Base Formula with a Calculator

Evaluate ${\mathrm{log}}_{2}\left(10\right)$ 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.

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

Try It

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

Solution

\frac{l n 100}{l n 5} \approx \frac{4.6051}{1.6094} = 2.861

$\frac{\mathrm{ln}100}{\mathrm{ln}5}\approx \frac{4.6051}{1.6094}=2.861$

Try it

The first graphing calculators were programmed to only handle logarithms with base 10. One clever way to create the graph of a logarithm with a different base was to change the base of the logarithm using the principles from this section.

In the graph below, you will see the graph of $f(x)=\frac{\log_{10}{x}}{\log_{10}{2}}$ .

Follow these steps to see a clever way to graph a logarithmic function with base other than 10 on a graphing tool that only knows base 10.

In the next line of the graph, enter the function $g(x) = \log_{2}{x}$
Can you tell the difference between the graph of this function and the graph of $f(x)$ ? Explain what you think is happening.
Your challenge is to write two new functions $h(x),\text{ and }k(x)$ that include a slider so you can change the base of the functions. Remember that there are restrictions on what values the base of a logarithm can take. You can click on the endpoints of the slider to change the input values.

https://www.desmos.com/calculator/umnz24xgl1

Exponential and Logarithmic Equations and Models

Learning Objectives

Use the change-of-base formula for logarithms

A General Note: The Change-of-Base Formula

How To: Given a logarithm with the form ${\mathrm{log}}_{b}M$ , use the change-of-base formula to rewrite it as a quotient of logs with any positive base $n$ , where $n\ne 1$ .

Example: Changing Logarithmic Expressions to Expressions Involving Only Natural Logs

Try It

Q & A

Can we change common logarithms to natural logarithms?

Example: Using the Change-of-Base Formula with a Calculator

Try It

Try it

Candela Citations

Exponential and Logarithmic Equations and Models

Change of Base

Learning Objectives

Use the change-of-base formula for logarithms

A General Note: The Change-of-Base Formula

How To: Given a logarithm with the form logbMlogbM{\mathrm{log}}_{b}M, use the change-of-base formula to rewrite it as a quotient of logs with any positive base nnn, where n≠1n≠1n\ne 1.

Example: Changing Logarithmic Expressions to Expressions Involving Only Natural Logs

Try It

Q & A

Can we change common logarithms to natural logarithms?

Example: Using the Change-of-Base Formula with a Calculator

Try It

Try it

How To: Given a logarithm with the form ${\mathrm{log}}_{b}M$ , use the change-of-base formula to rewrite it as a quotient of logs with any positive base $n$ , where $n\ne 1$ .