How to Evaluate Logarithms? (+FREE Worksheet!)

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

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. $$\color{blue}{log_{2}{32}}$$

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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