# 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.

##### Logarithmic Laws

Constant Values:

It can be automatically determined that:

• $log_a\left(1\right)=0$
• $log_a\left(a\right)=1$

Basic Arithmetic Operations

Multiplication, division, subtraction, and addition operations between logarithms:

• $log_aMN=log_aM+log_aN$
• $log_aM/N=log_aM-log_aN$
• $Log_a\left(M\right)\times Log_n\left(D\right)=Log_n\left(M\right)\times Log_a\left(D\right)$
• $Log_aM^n=nLog_aM$

Changing the Base of a Logarithm:

• $log_b\left(x\right)=log_c\left(x\right)/log_c\left(b\right)$
• $log_b\left(c\right)=1/log_c\left(b\right)$

Derivative of the Logarithm:

$fx=log_b\left(x\right)⇒f^{\prime}x=1/xln(b)$

Integral of the Logarithm:

$∫log_b\left(x\right)dx=x\times log_b\left(x\right)-1/ln\left(b\right)+C$

## Examples with solutions for Rules of Logarithms

### Exercise #1

$2\log_82+\log_83=$

### Step-by-Step Solution

$2\log_82=\log_82^2=\log_84$

$2\log_82+\log_83=\log_84+\log_83=$

$\log_84\cdot3=\log_812$

$\log_812$

### Exercise #2

$3\log_49+8\log_4\frac{1}{3}=$

### Step-by-Step Solution

Where:

$3\log_49=\log_49^3=\log_4729$

y

$8\log_4\frac{1}{3}=\log_4\left(\frac{1}{3}\right)^8=$

$\log_4\frac{1}{3^8}=\log_4\frac{1}{6561}$

Therefore

$3\log_49+8\log_4\frac{1}{3}=$

$\log_4729+\log_4\frac{1}{6561}$

$\log_ax+\log_ay=\log_axy$

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

$\log_49^{-1}=-\log_49$

$-\log_49$

### Exercise #3

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

### Step-by-Step Solution

We break it down into parts

$\log_61296=x$

$6^x=1296$

$x=4$

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

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

$\log_6\left(\frac{1}{2}:3\right)=\log_6\frac{1}{6}$

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

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

$x=-1$

$-1$

### Exercise #4

$\frac{1}{2}\log_24\times\log_38+\log_39\times\log_37=$

### Step-by-Step Solution

We break it down into parts

$\log_24=x$

$2^x=4$

$x=2$

$\log_39=x$

$3^x=9$

$x=2$

We substitute into the equation

$\frac{1}{2}\cdot2\log_38+2\log_37=$

$1\cdot\log_38+2\log_37=$

$\log_38+\log_37^2=$

$\log_38+\log_349=$

$\log_3\left(8\cdot49\right)=\log_3392$$x=2$

$\log_3392$

### Exercise #5

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

?=x

### Step-by-Step Solution

$\log_ax-\log_ay=\log_a\frac{x}{y}$

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

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

$7^3=\frac{x^2}{2}$

We multiply by: $2$

$2\cdot7^3=x^2$

Extract the root

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

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

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

### Exercise #6

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

$?=x$

### Step-by-Step Solution

Defined domain

x>0

x+1>0

x>-1

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

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

We reduce by: $7$ and by $X$

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

$x^2+x-2=0$

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

$x+2=0$

$x=-2$

Undefined domain x>0

$x-1=0$

$x=1$

Defined domain

$1$

### Exercise #7

$\log_23x\times\log_58=\log_5a+\log_52a$

Given a>0 , express X by a

### Step-by-Step Solution

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

### Exercise #8

$\log_{10}3+\log_{10}4=$

### Video Solution

$\log_{10}12$

### Exercise #9

$\log_49\times\log_{13}7=$

### Video Solution

$\log_{13}9\times\log_47$

### Exercise #10

$\log_29-\log_23=$

### Video Solution

$\log_23$

### Exercise #11

$\log_24+\log_25=$

### Video Solution

$\log_220$

### Exercise #12

$\log_53-\log_52=$

### Video Solution

$\log_51.5$

### Exercise #13

$2\log_38=$

### Video Solution

$\log_364$

### Exercise #14

$3\log_76=$

### Video Solution

$\log_7216$
$\log_75-\log_72=$
$\log_72.5$