Expanding and Condensing 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.