# How to Evaluate Logarithm

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.

## 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.

1. $$\color{blue}{3\log_{2}{64}}$$
2. $$\color{blue}{\frac{1}{2}\log_{6}{36}}$$
3. $$\color{blue}{\frac{1}{3}\log_{3}{27}}$$
4. $$\color{blue}{\log_{4}{64}}$$
5. $$\color{blue}{\log_{1000}{1}}$$
6. $$\color{blue}{\log_{620}{620}}$$
1. $$\color{blue}{18}$$
2. $$\color{blue}{1}$$
3. $$\color{blue}{1}$$
4. $$\color{blue}{3}$$
5. $$\color{blue}{0}$$
6. $$\color{blue}{1}$$

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