Since learning the rules of logarithms is essential for evaluating logarithms, this blog post will teach you some logarithmic rules for the convenience of your work in evaluating logarithms.

## Related Topics

- How to Solve Natural Logarithms
- How to Use Properties of Logarithms
- How to Solve Logarithmic Equations

## Necessary Logarithms Rules

- Logarithm is another way of writing exponent. \(\log_{b}{y}=x\) is equivalent to \(y=b^x\).
- Learn some logarithms rules: \((a>0,a≠0,M>0,N>0\), and k is a real number.)

Rule 1: \(\log_{a}{M.N} =\log_{a}{M} +\log_{a}{N}\)

Rule 2: \(\log_{a}{\frac{M}{N}}=\log_{a}{M} -\log_{a}{N} \)

Rule 3: \(\log_{a}{(M)^k} =k\log_{a}{M}\)

Rule 4: \(\log_{a}{a}=1\)

Rule 5:\(\log_{a}{1}=0\)

Rule 6: \(a^{\log_{a}{k}}=k\)

## Examples

### Evaluating Logarithm – Example 1:

Evaluate: \(\log_{2}{32}\)

**Solution:**

Rewrite 32 in power base form: \(32=2^5\), then:\(\log_{2}{32}=\log_{2}{(2)^5}\)

Use log rule:\(\log_{a}{(M)^{k}}=k.\log_{a}{M}→\log_{2}{(2)^5}=5\log_{2}{(2)}\)

Use log rule: \(\log_{a}{(a)}=1→\log_{2}{(2)} =1.\)

\(5\log_{2}{(2)}=5×1=5\)

### Evaluating Logarithm – Example 2:

Evaluate: \(3\log_{5}{125}\)

**Solution:**

Rewrite 125 in power base form: \(125=5^3\), then:

\(\log_{5}{125}=\log_{5}{(5)^3}\)

Use log rule: \(\log_{a}{(M)^k}=k.\log_{a}{M} →\log_{5}{(5)^3}=3\log_{5}{(5)}\)

Use log rule: \(\log_{a}{(a)} =1→ \log_{5}{(5)} =1.\)

\(3×3\log_{5}{(5)} =3×3=9\)

### Evaluating Logarithm – Example 3:

Evaluate: \(\log_{10}{1000}\)

**Solution:**

Rewrite 1000 in power base form: \(1000=10^3\), then:\(\log_{10}{1000}=\log_{10}{(10)^3}\)

Use log rule:\(\log_{a}{(M)^{k}}=k.\log_{a}{M}→\log_{10}{(10)^3}=3\log_{10}{(10)}\)

Use log rule: \(\log_{a}{(a)}=1→\log_{10}{(10)} =1.\)

\(3\log_{10}{(10)}=3×1=3\)

### Evaluating Logarithm – Example 4:

Evaluate: \(5\log_{3}{81}\)

**Solution:**

Rewrite 81 in power base form: \(81=3^4\), then:

\(\log_{3}{81}=\log_{3}{(3)^4}\)

Use log rule: \(\log_{a}{(M)^k}=k.\log_{a}{M} →\log_{3}{(3)^4}=4\log_{3}{(3)}\)

Use log rule: \(\log_{a}{(a)} =1→ \log_{3}{(3)} =1.\)

\(5×4\log_{3}{(3)} =5×4=20\)

## Exercises for Evaluating Logarithm

### Evaluate Logarithm.

- \(\color{blue}{3\log_{2}{64}}\)
- \(\color{blue}{\frac{1}{2}\log_{6}{36}}\)
- \(\color{blue}{\frac{1}{3}\log_{3}{27}}\)
- \(\color{blue}{\log_{4}{64}}\)
- \(\color{blue}{\log_{1000}{1}}\)
- \(\color{blue}{\log_{620}{620}}\)

- \(\color{blue}{18}\)
- \(\color{blue}{1}\)
- \(\color{blue}{1}\)
- \(\color{blue}{3}\)
- \(\color{blue}{0}\)
- \(\color{blue}{1}\)