How to Evaluate Logarithms

How to Evaluate Logarithms

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

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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: \(\log_{2}{16}=\log_{2}{(4^2)}\)

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

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: \(\log_{5}{625}=\log_{5}{(5^5)}\)

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

Answers

  1. -1
  2. -3
  3. -1
  4. -2
  5. 2
  6. 3
  7. 2
  8. 5

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