The Sum of Logarithms

Addition of Logarithms

The definition of a logarithm is:

$log_a⁡x=b$
$X=a^b$

Where:
$a$ is the base of the exponent
$X$ is what appears inside the log, can also appear in parentheses
$b$ is the exponent we raise the log base to in order to get the number that appears inside the log.

Adding logarithms with the same base is based on the following rule:

$log_a⁡x+log_a⁡y=log_a⁡(x\cdot y)$

Adding logarithms with different bases is done by changing the base of the log using the following rule:

$log_aX=\frac{log_{base~we~want~to~change~to}X}{log_{base~we~want~to~change~to}a}$

Practice The Sum of Logarithms

Question 1

$2\log_82+\log_83=$

Question 2

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

Question 3

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

Question 4

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

$$ ?=x $$

Question 5

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

Examples with solutions for The Sum of Logarithms

Exercise #1

$2\log_82+\log_83=$

Video Solution

Step-by-Step Solution

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

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

$\log_84\cdot3=\log_812$

Answer

$\log_812$

Exercise #2

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

Video Solution

Step-by-Step Solution

Where:

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

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

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

Therefore

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

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

$\log_ax+\log_ay=\log_axy$

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

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

Answer

$-\log_49$

Exercise #3

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

Video Solution

Step-by-Step Solution

We break it down into parts

$\log_24=x$

$2^x=4$

$x=2$

$\log_39=x$

$3^x=9$

$x=2$

We substitute into the equation

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

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

$\log_38+\log_37^2=$

$\log_38+\log_349=$

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

Answer

$\log_3392$

Exercise #4

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

$?=x$

Video Solution

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

Answer

$1$

Exercise #5

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

Video Solution

Answer

$\log_{10}12$

Question 1

$\log_24+\log_25=$

Question 2

$\log_974+\log_9\frac{1}{2}=$

Question 3

$\log_2x+\log_2\frac{x}{2}=5$

?=x

Question 4

$\log_4x+\log_4(x+2)=2$

Question 5

?=a

$\ln(a+5)+\ln(a+7)=0$

Exercise #6

$\log_24+\log_25=$

Video Solution

Answer

$\log_220$

Exercise #7

$\log_974+\log_9\frac{1}{2}=$

Video Solution

Answer

$\log_937$

Exercise #8

$\log_2x+\log_2\frac{x}{2}=5$

?=x

Video Solution

Answer

$8$

Exercise #9

$\log_4x+\log_4(x+2)=2$

Video Solution

Answer

$-1+\sqrt{17}$

Exercise #10

?=a

$\ln(a+5)+\ln(a+7)=0$

Video Solution

Answer

$-6+\sqrt{2}$

Question 1

$\log3x+\log(x-1)=3$

$$ ?=x $$

Question 2

Find X

$\frac{\log_84x+\log_8(x+2)}{\log_83}=3$

Question 3

$\log4x+\log2-\log9=\log_24$

?=x

Question 4

$\log_9e^3\times(\log_224-\log_28)(\ln8+\ln2)$

Question 5

$\frac{\log_45+\log_42}{3\log_42}=$

Exercise #11

$\log3x+\log(x-1)=3$

$?=x$

Video Solution

Answer

$18.8$

Exercise #12

Find X

$\frac{\log_84x+\log_8(x+2)}{\log_83}=3$

Video Solution

Answer

$-1+\frac{\sqrt{31}}{2}$

Exercise #13

$\log4x+\log2-\log9=\log_24$

?=x

Video Solution

Answer

$112.5$

Exercise #14

$\log_9e^3\times(\log_224-\log_28)(\ln8+\ln2)$

Video Solution

Answer

$6$

Exercise #15

$\frac{\log_45+\log_42}{3\log_42}=$

Video Solution

Answer

$\log_810$

The Sum of Logarithms - Examples, Exercises and Solutions

Addition of Logarithms

The definition of a logarithm is:

Adding logarithms with the same base is based on the following rule:

Adding logarithms with different bases is done by changing the base of the log using the following rule:

Practice The Sum of Logarithms

Examples with solutions for The Sum of Logarithms

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

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Exercise #3

Video Solution

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Exercise #4

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Exercise #5

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Exercise #6

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Exercise #7

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Exercise #8

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Exercise #9

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Exercise #10

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Exercise #11

Video Solution

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Exercise #12

Video Solution

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Exercise #13

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Exercise #14

Video Solution

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Exercise #15

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