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Study Guides > College Algebra

Evaluate Logarithms

Learning Objectives

  • Evaluate logarithms with and without a calculator
  • Evaluate logarithms with base 10, and base e
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: Solving Logarithms Mentally

Solve y=log4(64)y={\mathrm{log}}_{4}\left(64\right) without using a calculator.

Answer: 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

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

Answer: log121(11)=12{\mathrm{log}}_{121}\left(11\right)=\frac{1}{2} (recalling that 121=(121)12=11\sqrt{121}={\left(121\right)}^{\frac{1}{2}}=11 )

Example: Evaluating the Logarithm of a Reciprocal

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

Answer: 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 b−a=1ba{b}^{-a}=\frac{1}{{b}^{a}}. We use this information to write

3−3=133=127\begin{array}{l}{3}^{-3}=\frac{1}{{3}^{3}}\hfill \\ =\frac{1}{27}\hfill \end{array}

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

Try It

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

Answer: log2(132)=−5{\mathrm{log}}_{2}\left(\frac{1}{32}\right)=-5

Use common logarithms

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base b, exponent x, and output y. Then we write x=logb(y)x={\mathrm{log}}_{b}\left(y\right).

Example: Converting from Exponential Form to Logarithmic Form

Write the following exponential equations in logarithmic form.
  1. 23=8{2}^{3}=8
  2. 52=25{5}^{2}=25
  3. 10−4=110,000{10}^{-4}=\frac{1}{10,000}

Answer: First, identify the values of b, y, and x. Then, write the equation in the form x=logb(y)x={\mathrm{log}}_{b}\left(y\right).

  1. 23=8{2}^{3}=8Here, b = 2, x = 3, and y = 8. Therefore, the equation 23=8{2}^{3}=8 is equivalent to log2(8)=3{\mathrm{log}}_{2}\left(8\right)=3.
  2. 52=25{5}^{2}=25Here, b = 5, x = 2, and y = 25. Therefore, the equation 52=25{5}^{2}=25 is equivalent to log5(25)=2{\mathrm{log}}_{5}\left(25\right)=2.
  3. 10−4=110,000{10}^{-4}=\frac{1}{10,000}Here, b = 10, x = –4, and y=110,000y=\frac{1}{10,000}. Therefore, the equation 10−4=110,000{10}^{-4}=\frac{1}{10,000} is equivalent to log10(110,000)=−4{\text{log}}_{10}\left(\frac{1}{10,000}\right)=-4.

Try It

Write the following exponential equations in logarithmic form.
  1. 32=9{3}^{2}=9
  2. 53=125{5}^{3}=125
  3. 2−1=12{2}^{-1}=\frac{1}{2}

Answer:

  1. 32=9{3}^{2}=9 is equivalent to log3(9)=2{\mathrm{log}}_{3}\left(9\right)=2
  2. 53=125{5}^{3}=125 is equivalent to log5(125)=3{\mathrm{log}}_{5}\left(125\right)=3
  3. 2−1=12{2}^{-1}=\frac{1}{2} is equivalent to log2(12)=−1{\text{log}}_{2}\left(\frac{1}{2}\right)=-1

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