How to Evaluate Logarithms? (+FREE Worksheet!)

Do you want to know how to evaluate Logarithms? You can do it in a few easy steps!

How to Evaluate Logarithms? (+FREE Worksheet!)

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Step by step guide to Evaluating Logarithms

  • Logarithm is another way of writing exponent. \(log_{b}{y}=x\) is equivalent to \(y=b^x \)
  • Learn some logarithms rules:

\(log_{b}{(x)}=\frac{log_{d}{(x)}}{\log_{d}{(b)}}\)
\(log_{a}{x^b}=b log_{a}{x}\)
\(log_{a}{1}=0\)
\(log_{a}{a}=1\)

Evaluating logarithms – Example 1:

Evaluate: \(log_{2}{16}\)

Solution:

Rewrite \(16\)  in power base form: \(16=2^4\) , then: \(log_{2}{16}=log_{2}{(2^4)}\)

Use log rule: \(log_{a}{x^b}=b log_{a}{x}\), then: \(log_{2}{(2^4)}=4log_{2}{2}\)
Use log rule: \(log_{a}{a}=1\), then: \( 4log_{2}{2}=4\times1=4\)

Evaluating logarithms – Example 2:

Evaluate: \(log_{6}{216}\)

Solution:

Rewrite \(216\) in power base form: \(216=6^3\) , then: \(log_{6}{216}=log_{6}{(6^3)}\)

Use log rule: \(log_{a}{x^b}=b log_{a}{x}\), then: \(log_{6}{(6^3)}=3 log_{6}{6}\)
Use log rule: \(log_{a}{a}=1\), then: \( 3 log_{6}{6}=3\times1=3\)

Evaluating logarithms – Example 3:

Evaluate: \(log_{4}{64}\)

Solution:

Rewrite \(64\) in power base form: \(64=4^3\) , then: \(log_{4}{64}=log_{4}{(4^3)}\)

Use log rule: \( log_{a}{x^b}=b log_{a}{x}\), then: \(log_{4}{(4^3)}=3 log_{4}{4}\)
Use log rule: \(log_{a}{a}=1\), then: \( 3 log_{4}{4}=3\times1=3\)

Evaluating logarithms – Example 4:

Evaluate: \(log_{5}{625}\)

Solution:

Rewrite \(625\) in power base form: \(625=5^4\) , then: \(log_{5}{625}=log_{5}{(5^4)}\)

Use log rule: \( log_{a}{x^b}=b log_{a}{x}\), then: \(log_{5}{(5^4)}=4 log_{5}{5}\)
Use log rule: \(log_{a}{a}=1\), then: \(4 log_{5}{5}=4\times1=4\)

Evaluating logarithms Exercises

Evaluate each logarithm.

  1. \(\color{blue}{log_{2}{\frac{1}{2}}}\)
  2. \(\color{blue}{log_{2}{\frac{1}{8}}}\)
  3. \(\color{blue}{log_{3}{\frac{1}{3}}}\)
  4. \(\color{blue}{log_{4}{\frac{1}{16}}}\)
  5. \(\color{blue}{log_{5}{25}}\)
  6. \(\color{blue}{log_{3}{27}}\)
  7. \(\color{blue}{log_{3}{9}}\)
  8. \(\color{blue}{log_{2}{32}}\)

Answers

  1. \(\color{blue}{-1}\)
  2. \(\color{blue}{-3}\)
  3. \(\color{blue}{-1}\)
  4. \(\color{blue}{-2}\)
  5. \(\color{blue}{2}\)
  6. \(\color{blue}{3}\)
  7. \(\color{blue}{2}\)
  8. \(\color{blue}{5}\)

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Math and Critical Thinking Challenges: For the Middle and High School Student