Using 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 \neq 1 and b \neq 1, we show

\log _{b} M=\frac{\log _{n} M}{\log _{n} b}

Let y=log_bM. By exponentiating both sides with base b, we arrive at an exponential form, namely b^y=M. It follows that

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

For example, to evaluate log_536 using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.

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


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 \neq 1 and b \neq 1,

\log _{b} M=\frac{\log _{n} M}{\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.

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

and

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


HOW TO

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

  1. Determine the new base n, remembering that the common \log , \log (x), has base 10 , and the natural \log , \ln (x), has base e.
  2. Rewrite the log as a quotient using the change-of-base formula
    1. The numerator of the quotient will be a logarithm with base n and argument M.
    2. 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 log_53 to a quotient of natural logarithms.


Solution

Because we will be expressing \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{aligned} \log _{b} M &=\frac{\ln M}{\ln b} \\ \log _{5} 3 &=\frac{\ln 3}{\ln 5} \end{aligned}


TRY IT #13

Change log_{0.5}8 to a quotient of natural logarithms.


Q&A

Can we change common logarithms to natural logarithms?

Yes. Remember that log9 means log_{10}9. So, log9= \frac{\ln9}{\ln10}.


EXAMPLE 14

Using the Change-of-Base Formula with a Calculator

Evaluate log_2(10) 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{aligned} &\log _{2} 10=\frac{\ln 10}{\ln 2} & & \text { Apply the change of base formula using base e }. \\ &\approx 3.3219 & & \text { Use a calculator to evaluate to 4 decimal places. } \end{aligned}


TRY IT #14

Evaluate log_5(100) using the change-of-base formula.