Evaluate logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider [latex]{\mathrm{log}}_{2}8\\[/latex]. We ask, “To what exponent must 2 be raised in order to get 8?” Because we already know [latex]{2}^{3}=8\\[/latex], it follows that [latex]{\mathrm{log}}_{2}8=3\\[/latex].

Now consider solving [latex]{\mathrm{log}}_{7}49\\[/latex] and [latex]{\mathrm{log}}_{3}27\\[/latex] mentally.

  • We ask, “To what exponent must 7 be raised in order to get 49?” We know [latex]{7}^{2}=49\\[/latex]. Therefore, [latex]{\mathrm{log}}_{7}49=2\\[/latex]
  • We ask, “To what exponent must 3 be raised in order to get 27?” We know [latex]{3}^{3}=27\\[/latex]. Therefore, [latex]{\mathrm{log}}_{3}27=3\\[/latex]

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate [latex]{\mathrm{log}}_{\frac{2}{3}}\frac{4}{9}\\[/latex] mentally.

  • We ask, “To what exponent must [latex]\frac{2}{3}\\[/latex] be raised in order to get [latex]\frac{4}{9}\\[/latex]? ” We know [latex]{2}^{2}=4\\[/latex] and [latex]{3}^{2}=9\\[/latex], so [latex]{\left(\frac{2}{3}\right)}^{2}=\frac{4}{9}\\[/latex]. Therefore, [latex]{\mathrm{log}}_{\frac{2}{3}}\left(\frac{4}{9}\right)=2\\[/latex].

How To: Given a logarithm of the form [latex]y={\mathrm{log}}_{b}\left(x\right)\\[/latex], evaluate it mentally.

  1. Rewrite the argument x as a power of b: [latex]{b}^{y}=x\\[/latex].
  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 [latex]y={\mathrm{log}}_{4}\left(64\right)\\[/latex] without using a calculator.

Solution

First we rewrite the logarithm in exponential form: [latex]{4}^{y}=64\\[/latex]. Next, we ask, “To what exponent must 4 be raised in order to get 64?”

We know

[latex]{4}^{3}=64\\[/latex]

Therefore,

[latex]\mathrm{log}{}_{4}\left(64\right)=3\\[/latex]

Try It 3

Solve [latex]y={\mathrm{log}}_{121}\left(11\right)\\[/latex] without using a calculator.

Solution

Example 4: Evaluating the Logarithm of a Reciprocal

Evaluate [latex]y={\mathrm{log}}_{3}\left(\frac{1}{27}\right)\\[/latex] without using a calculator.

Solution

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

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

[latex]\begin{cases}{3}^{-3}=\frac{1}{{3}^{3}}\hfill \\ =\frac{1}{27}\hfill \end{cases}\\[/latex]

Therefore, [latex]{\mathrm{log}}_{3}\left(\frac{1}{27}\right)=-3\\[/latex].

Try It 4

Evaluate [latex]y={\mathrm{log}}_{2}\left(\frac{1}{32}\right)\\[/latex] without using a calculator.

Solution