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: \(\log_{2}{16}\)

**Answer:**

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: \(\log_{6}{216}\)

**Answer:**

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: \(\log_{4}{64}\)

**Answer:**

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: \(\log_{5}{625}\)

**Answer:**

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.

- \(\log_{2}{\frac{1}{2}}\)
- \(\log_{2}{\frac{1}{8}}\)
- \(\log_{3}{\frac{1}{3}}\)
- \(\log_{4}{\frac{1}{16}}\)
- \(\log_{5}{25}\)
- \(\log_{3}{27}\)
- \(\log_{3}{9}\)
- \(\log_{2}{32}\)

## Answers

- -1
- -3
- -1
- -2
- 2
- 3
- 2
- 5