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Study Guides > College Algebra

Evaluate logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider log28{\mathrm{log}}_{2}8. We ask, "To what exponent must 2 be raised in order to get 8?" Because we already know 23=8{2}^{3}=8, it follows that log28=3{\mathrm{log}}_{2}8=3.

Now consider solving log749{\mathrm{log}}_{7}49 and log327{\mathrm{log}}_{3}27 mentally.

  • We ask, "To what exponent must 7 be raised in order to get 49?" We know 72=49{7}^{2}=49. Therefore, log749=2{\mathrm{log}}_{7}49=2
  • We ask, "To what exponent must 3 be raised in order to get 27?" We know 33=27{3}^{3}=27. Therefore, log327=3{\mathrm{log}}_{3}27=3

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate log2349{\mathrm{log}}_{\frac{2}{3}}\frac{4}{9} mentally.

  • We ask, "To what exponent must 23\frac{2}{3} be raised in order to get 49\frac{4}{9}? " We know 22=4{2}^{2}=4 and 32=9{3}^{2}=9, so (23)2=49{\left(\frac{2}{3}\right)}^{2}=\frac{4}{9}. Therefore, log23(49)=2{\mathrm{log}}_{\frac{2}{3}}\left(\frac{4}{9}\right)=2.

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

  1. Rewrite the argument x as a power of b: by=x{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=log4(64)y={\mathrm{log}}_{4}\left(64\right) without using a calculator.

Solution

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

We know

43=64{4}^{3}=64

Therefore,

log4(64)=3\mathrm{log}{}_{4}\left(64\right)=3

Try It 3

Solve y=log121(11)y={\mathrm{log}}_{121}\left(11\right) without using a calculator.

Solution

Example 4: Evaluating the Logarithm of a Reciprocal

Evaluate y=log3(127)y={\mathrm{log}}_{3}\left(\frac{1}{27}\right) without using a calculator.

Solution

First we rewrite the logarithm in exponential form: 3y=127{3}^{y}=\frac{1}{27}. Next, we ask, "To what exponent must 3 be raised in order to get 127\frac{1}{27}"?

We know 33=27{3}^{3}=27, but what must we do to get the reciprocal, 127\frac{1}{27}? Recall from working with exponents that ba=1ba{b}^{-a}=\frac{1}{{b}^{a}}. We use this information to write

{33=133=127\begin{cases}{3}^{-3}=\frac{1}{{3}^{3}}\hfill \\ =\frac{1}{27}\hfill \end{cases}

Therefore, log3(127)=3{\mathrm{log}}_{3}\left(\frac{1}{27}\right)=-3.

Try It 4

Evaluate y=log2(132)y={\mathrm{log}}_{2}\left(\frac{1}{32}\right) without using a calculator.

Solution

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