How to Evaluate Logarithms? (+FREE Worksheet!)
Evaluate Logarithms: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Translate firstAsk: the base to what power gives the input?
- Use rules legallyProducts, quotients, and powers have rules; sums do not split.
- Protect the domainKeep the log input positive and track asymptotes when graphing.
Worked examples
Evaluate a log
- Ask 3 to what power equals 81.
- 3 to the fourth power is 81.
- The logarithm is that exponent.
Find a log domain
- The input is x – 5.
- Require x – 5 > 0.
- Solve the inequality.
Try one before moving on
Evaluate Logarithms: pop-up practice
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: \( 4log_{2}{2}=4\times1=4\)
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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.
- \(\color{blue}{log_{2}{\frac{1}{2}}}\)
- \(\color{blue}{log_{2}{\frac{1}{8}}}\)
- \(\color{blue}{log_{3}{\frac{1}{3}}}\)
- \(\color{blue}{log_{4}{\frac{1}{16}}}\)
- \(\color{blue}{log_{5}{25}}\)
- \(\color{blue}{log_{3}{27}}\)
- \(\color{blue}{log_{3}{9}}\)
- \(\color{blue}{log_{2}{32}}\)
Answers
- \(\color{blue}{-1}\)
- \(\color{blue}{-3}\)
- \(\color{blue}{-1}\)
- \(\color{blue}{-2}\)
- \(\color{blue}{2}\)
- \(\color{blue}{3}\)
- \(\color{blue}{2}\)
- \(\color{blue}{5}\)
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