# How to Evaluate Logarithms? (+FREE Worksheet!)

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

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## 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.

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

## 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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