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

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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