# Evaluating Logarithms

Do you want to know how to solve Evaluating Logarithms? you can do it in two 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:

$$\log_{b}{(x)}=\frac{\log_{d}{(x)}}{\log_{d}{(b)}}$$
$$\log_{a}{x^b}=b\log_{a}{x}$$
$$\log_{a}{1}=0$$
$$\log_{a}{a}=1$$

### Example 1:

Evaluate: $$\log_{2}{16}$$

Solution:

Rewrite 16 in power base form: $$16=2^4$$ , then: $$\log_{2}{16}=\log_{2}{(4^2)}$$

Use log rule: $$\log_{a}{x^b}=b\log_{a}{x}$$, then: $$\log_{2}{(4^2)}=4\log_{2}{2}$$
Use log rule: $$\log_{a}{a}=1$$, then: $$4\log_{2}{2}=4\times1=4$$

### 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: $$3\log_{6}{6}=3\times1=3$$

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

### Example 4:

Evaluate: $$\log_{5}{625}$$

Solution:

Rewrite 625 in power base form: $$625=5^4$$ , then: $$\log_{5}{625}=\log_{5}{(5^5)}$$

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: $$4\log_{5}{5}=4\times1=4$$

## Exercises

### Evaluate each logarithm.

1. $$\log_{2}{\frac{1}{2}}$$
2. $$\log_{2}{\frac{1}{8}}$$
3. $$\log_{3}{\frac{1}{3}}$$
4. $$\log_{4}{\frac{1}{16}}$$
5. $$\log_{5}{25}$$
6. $$\log_{3}{27}$$
7. $$\log_{3}{9}$$
8. $$\log_{2}{32}$$

1. -1
2. -3
3. -1
4. -2
5. 2
6. 3
7. 2
8. 5

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