How to Evaluate Logarithms? (+FREE Worksheet!)
How to evaluate logarithms, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.
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}=4times1=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}=3times1=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}=3times1=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}=4times1=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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