Evaluating Logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider \(log_28\). We ask, "To what exponent must \(2\) be raised in order to get 8?" Because we already know \(2^3=8\), it follows that \(log_28=3\).

Now consider solving \(log_749\) and \(log_327\) mentally.

  • We ask, "To what exponent must 7 be raised in order to get 49?" We know \(7^2=49\). Therefore, \(log_749=2\)
  • We ask, "To what exponent must 3 be raised in order to get 27?" We know \(3^3=27\). Therefore, \(log_327=3\)

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let's evaluate \(log_{\frac{2}{3}} \frac{4}{9}\) mentally.

  • We ask, "To what exponent must \(\frac{2}{3}\) be raised in order to get \(\frac{4}{9}\)? " We know \(2^2=4\) and \(3^2=9\), so \((\frac{2}{3})^2= \frac{4}{9}\). Therefore, \(log_{\frac{2}{3}}(\frac{4}{9})=2\).


HOW TO

Given a logarithm of the form \(y=log_b(x)\), evaluate it mentally.

  1. Rewrite the argument \(x\) as a power of \(b: b^y=x\).
  2. Use previous knowledge of powers of \(b\) identify \(y\) by asking, "To what exponent should \(b\) be raised in order to get \(x\)?"


EXAMPLE 3

Solving Logarithms Mentally

Solve \(y=log_4(64)\) without using a calculator.


Solution

First we rewrite the logarithm in exponential form: \(4^y=64\). Next, we ask, "To what exponent must 4 be raised in order to get 64?"

We know

\(4^3=64\)

Therefore,

\(log_4(64)=3\)


TRY IT #3

Solve \(y=log_{121}\) (11) without using a calculator.


EXAMPLE 4

Evaluating the Logarithm of a Reciprocal

Evaluate \(y=log_3(\frac{1}{27})\) without using a calculator.


Solution

First we rewrite the logarithm in exponential form: \(3^y= \frac{1}{27}\). Next, we ask, "To what exponent must 3 be raised in order to get \(\frac{1}{27}\)?"

We know \(3^3=27\), but what must we do to get the reciprocal, \(\frac{1}{27}\)? Recall from working with exponents that \(b^{−a}= \frac{1}{b^a}\). We use this information to write

\(3^{−3}=\frac{1}{3^3}\)

\(=\frac{1}{27}\)

Therefore, \(log_3(\frac{1}{27})=−3\).


TRY IT #4

Evaluate \(y=log_2(\frac{1}{32})\) without using a calculator.