How to Evaluate Logarithms? (+FREE Worksheet!)

Related Topics

• How to Solve Natural Logarithms Problems
• How to Solve Logarithmic Equations
• Logarithms Properties

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: $4log_{2}{2}=4\times1=4$

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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: $3 log_{4}{4}=3\times1=3$

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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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