Rules of Logarithms - Examples, Exercises and Solutions

What Are Logarithmic Laws?

There are a few logarithmic laws worth knowing to make solving problems easier. The following laws are the main rules you will use. It should be noted that the letters a, m, n must be positive real numbers for these laws to be valid.

Practice Rules of Logarithms

Exercise #1

2log82+log83= 2\log_82+\log_83=

Video Solution

Step-by-Step Solution

2log82=log822=log84 2\log_82=\log_82^2=\log_84

2log82+log83=log84+log83= 2\log_82+\log_83=\log_84+\log_83=

log843=log812 \log_84\cdot3=\log_812

Answer

log812 \log_812

Exercise #2

3log49+8log413= 3\log_49+8\log_4\frac{1}{3}=

Video Solution

Step-by-Step Solution

Where:

3log49=log493=log4729 3\log_49=\log_49^3=\log_4729

y

8log413=log4(13)8= 8\log_4\frac{1}{3}=\log_4\left(\frac{1}{3}\right)^8=

log4138=log416561 \log_4\frac{1}{3^8}=\log_4\frac{1}{6561}

Therefore

3log49+8log413= 3\log_49+8\log_4\frac{1}{3}=

log4729+log416561 \log_4729+\log_4\frac{1}{6561}

logax+logay=logaxy \log_ax+\log_ay=\log_axy

(72916561)=log419 \left(729\cdot\frac{1}{6561}\right)=\log_4\frac{1}{9}

log491=log49 \log_49^{-1}=-\log_49

Answer

log49 -\log_49

Exercise #3

12log24×log38+log39×log37= \frac{1}{2}\log_24\times\log_38+\log_39\times\log_37=

Video Solution

Step-by-Step Solution

We break it down into parts

log24=x \log_24=x

2x=4 2^x=4

x=2 x=2

log39=x \log_39=x

3x=9 3^x=9

x=2 x=2

We substitute into the equation

122log38+2log37= \frac{1}{2}\cdot2\log_38+2\log_37=

1log38+2log37= 1\cdot\log_38+2\log_37=

log38+log372= \log_38+\log_37^2=

log38+log349= \log_38+\log_349=

log3(849)=log3392 \log_3\left(8\cdot49\right)=\log_3392 x=2 x=2

Answer

log3392 \log_3392

Exercise #4

14log61296log612log63= \frac{1}{4}\cdot\log_61296\cdot\log_6\frac{1}{2}-\log_63=

Video Solution

Step-by-Step Solution

We break it down into parts

log61296=x \log_61296=x

6x=1296 6^x=1296

x=4 x=4

144log612log63= \frac{1}{4}\cdot4\cdot\log_6\frac{1}{2}-\log_63=

log612log63= \log_6\frac{1}{2}-\log_63=

log6(12:3)=log616 \log_6\left(\frac{1}{2}:3\right)=\log_6\frac{1}{6}

log616=x \log_6\frac{1}{6}=x

6x=16 6^x=\frac{1}{6}

x=1 x=-1

Answer

1 -1

Exercise #5

log7x4log72x2=3 \log_7x^4-\log_72x^2=3

?=x

Video Solution

Step-by-Step Solution

logaxlogay=logaxy \log_ax-\log_ay=\log_a\frac{x}{y}

log7x4log72x2= \log_7x^4-\log_72x^2=

log7x42x2=3 \log_7\frac{x^4}{2x^2}=3

73=x22 7^3=\frac{x^2}{2}

We multiply by: 2 2

273=x2 2\cdot7^3=x^2

Extract the root

x=680=714 x=\sqrt{680}=7\sqrt{14}

x=680=714 x=-\sqrt{680}=-7\sqrt{14}

Answer

714  , 714 -7\sqrt{14\text{ }}\text{ , }7\sqrt{14}

Exercise #1

log7x+log(x+1)log7=log2xlogx \log7x+\log(x+1)-\log7=\log2x-\log x

?=x ?=x

Video Solution

Step-by-Step Solution

Defined domain

x>0

x+1>0

x>-1

log7x+log(x+1)log7=log2xlogx \log7x+\log\left(x+1\right)-\log7=\log2x-\log x

log7x(x+1)7=log2xx \log\frac{7x\cdot\left(x+1\right)}{7}=\log\frac{2x}{x}

We reduce by: 7 7 and by X X

x(x+1)=2 x\left(x+1\right)=2

x2+x2=0 x^2+x-2=0

(x+2)(x1)=0 \left(x+2\right)\left(x-1\right)=0

x+2=0 x+2=0

x=2 x=-2

Undefined domain x>0

x1=0 x-1=0

x=1 x=1

Defined domain

Answer

1 1

Exercise #2

log23x×log58=log5a+log52a \log_23x\times\log_58=\log_5a+\log_52a

Given a>0 , express X by a

Video Solution

Step-by-Step Solution

Answer

2a2273 \sqrt[3]{\frac{2a^2}{27}}

Exercise #3

log103+log104= \log_{10}3+\log_{10}4=

Video Solution

Answer

log1012 \log_{10}12

Exercise #4

log24+log25= \log_24+\log_25=

Video Solution

Answer

log220 \log_220

Exercise #5

log974+log912= \log_974+\log_9\frac{1}{2}=

Video Solution

Answer

log937 \log_937

Exercise #1

log53log52= \log_53-\log_52=

Video Solution

Answer

log51.5 \log_51.5

Exercise #2

log29log23= \log_29-\log_23=

Video Solution

Answer

log23 \log_23

Exercise #3

log75log72= \log_75-\log_72=

Video Solution

Answer

log72.5 \log_72.5

Exercise #4

log49×log137= \log_49\times\log_{13}7=

Video Solution

Answer

log139×log47 \log_{13}9\times\log_47

Exercise #5

logmn×logzr= \log_mn\times\log_zr=

Video Solution

Answer

logzn×logmr \log_zn\times\log_mr