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

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

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

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{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}

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

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}

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

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 $b\ne 1$ ,

{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}

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

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 13: 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{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}

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

Try It 13

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

Solution

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 14: 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{matrix} {l o g}_{2} 10 = \frac{l n 10}{l n 2} & Apply the change of base formula using base e . \approx 3.3219 & Use a calculator to evaluate to 4 decimal places . \end{matrix}

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

Try It 14

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

Solution

Logarithmic Properties