How to Evaluate Logarithm? (+FREE Worksheet!)

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.

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. $\color{blue}{1}$

The Absolute Best Book for the Algebra Test

Algebra I Practice Workbook The Most Comprehensive Review of Algebra 1

Download

$29.99 Original price was: $29.99.$19.99Current price is: $19.99.

Algebra II Practice Workbook The Most Comprehensive Review of Algebra 2

Download

$25.99 Original price was: $25.99.$19.99Current price is: $19.99.

High School Algebra I A Comprehensive Review and Step-by-Step Guide to Mastering High School Algebra 1

Download

$29.99 Original price was: $29.99.$19.99Current price is: $19.99.

Algebra II for Beginners The Ultimate Step by Step Guide to Acing Algebra II

Download

$29.99 Original price was: $29.99.$14.99Current price is: $14.99.

Rated 5.00 out of 5 based on 1 customer rating

Satisfied 1 Students

Pre-Algebra for Beginners The Ultimate Step by Step Guide to Preparing for the Pre-Algebra Test

Download

$24.99 Original price was: $24.99.$14.99Current price is: $14.99.

Rated 4.30 out of 5 based on 105 customer ratings

Satisfied 92 Students