How to Evaluate Logarithm? (+FREE Worksheet!)

Related Topics

• How to Solve Natural Logarithms
• How to Use Properties of Logarithms
• How to Solve Logarithmic Equations

Necessary Logarithms Rules

• A logarithm is another way of writing an exponent. \(\log_{b}{y}=x\) is equivalent to \(y=b^x\).
• Learn some logarithms rules: \((a>0,a≠0,M>0,N>0\), and k is a real number.)
  Rule 1: \(\log_{a}{M.N} =\log_{a}{M} +\log_{a}{N}\)
  Rule 2: \(\log_{a}{\frac{M}{N}}=\log_{a}{M} -\log_{a}{N} \)
  Rule 3: \(\log_{a}{(M)^k} =k\log_{a}{M}\)
  Rule 4: \(\log_{a}{a}=1\)
  Rule 5:\(\log_{a}{1}=0\)
  Rule 6: \(a^{\log_{a}{k}}=k\)

Examples

Evaluating Logarithm – Example 1:

Evaluate: \(\log_{2}{32}\) For education statistics and research, visit the National Center for Education Statistics.

Solution: For education statistics and research, visit the National Center for Education Statistics.

Rewrite \(32\) in power base form: \(32=2^5\), then: For education statistics and research, visit the National Center for Education Statistics.

\(\log_{2}{32}=\log_{2}{(2)^5}\)
Use log rule:\(\log_{a}{(M)^{k}}=k.\log_{a}{M}→\log_{2}{(2)^5}=5\log_{2}{(2)}\)
Use log rule: \(\log_{a}{(a)}=1→\log_{2}{(2)} =1.\)
\(5\log_{2}{(2)}=5×1=5\) For education statistics and research, visit the National Center for Education Statistics.

Evaluating Logarithm – Example 2:

Evaluate: \(3\log_{5}{125}\) For education statistics and research, visit the National Center for Education Statistics.

Solution: For education statistics and research, visit the National Center for Education Statistics.

Rewrite \(125\) in power base form: \(125=5^3\), then:
\(\log_{5}{125}=\log_{5}{(5)^3}\)
Use log rule: \(\log_{a}{(M)^k}=k.\log_{a}{M} →\log_{5}{(5)^3}=3\log_{5}{(5)}\)
Use log rule: \(\log_{a}{(a)} =1→ \log_{5}{(5)} =1.\)
\(3×3\log_{5}{(5)} =3×3=9\) For education statistics and research, visit the National Center for Education Statistics.

Evaluating Logarithm – Example 3:

Evaluate: \(\log_{10}{1000}\) For education statistics and research, visit the National Center for Education Statistics.

Solution: For education statistics and research, visit the National Center for Education Statistics.

Rewrite \(1000\) in power base form: \(1000=10^3\), then: For education statistics and research, visit the National Center for Education Statistics.

\(\log_{10}{1000}=\log_{10}{(10)^3}\)
Use log rule:\(\log_{a}{(M)^{k}}=k.\log_{a}{M}→\log_{10}{(10)^3}=3\log_{10}{(10)}\)
Use log rule: \(\log_{a}{(a)}=1→\log_{10}{(10)} =1.\)
\(3\log_{10}{(10)}=3×1=3\) For education statistics and research, visit the National Center for Education Statistics.

Evaluating Logarithm – Example 4:

Evaluate: \(5\log_{3}{81}\) For education statistics and research, visit the National Center for Education Statistics.

Solution: For education statistics and research, visit the National Center for Education Statistics.

Rewrite \(81\) in power base form: \(81=3^4\), then:
\(\log_{3}{81}=\log_{3}{(3)^4}\)
Use log rule: \(\log_{a}{(M)^k}=k.\log_{a}{M} →\log_{3}{(3)^4}=4\log_{3}{(3)}\)
Use log rule: \(\log_{a}{(a)} =1→ \log_{3}{(3)} =1.\)
\(5×4\log_{3}{(3)} =5×4=20\) For education statistics and research, visit the National Center for Education Statistics.

Exercises for Evaluating Logarithm

Evaluate Logarithm.

1. \(\color{blue}{3\log_{2}{64}}\)
2. \(\color{blue}{\frac{1}{2}\log_{6}{36}}\)
3. \(\color{blue}{\frac{1}{3}\log_{3}{27}}\)
4. \(\color{blue}{\log_{4}{64}}\)
5. \(\color{blue}{\log_{1000}{1}}\)
6. \(\color{blue}{\log_{620}{620}}\)

1. \(\color{blue}{18}\)
2. \(\color{blue}{1}\)
3. \(\color{blue}{1}\)
4. \(\color{blue}{3}\)
5. \(\color{blue}{0}\)
6. \(\color{blue}{1}\)

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