How To Solve Logarithmic Equations

How To Solve Logarthmic Equations Video

As the video above points out, there are two main types of logarithmic equations. Before you to decide how to solve an equation, you must determine whether the equation is has a logarithm on one side and a number on the other or whether it has logarithms on both sides.

Logarithm on one side and a number on the other

• log₄x+ log₄8=3

How to solve this kind (log on one side),

just rewrite the logarithm as exponential equation and solve.

Example of how to solve this kind of log equation

Step 1) Rewrite log side as single logarithm	log48x=3
Step2) Rewrite as exponential equation	43 = 8x
Step 3) Solve the exponential equation	64 =8x 8 = x

Logarithm on both sides

• log₅x + log2₃= log₅6

How to solve this kind (logarithm on both sides),

Step 1) use the rules of logarithms to rewrite the left side and the rigth side of the equation to a single logarithm
Step 2) "cancel" the log
Step 3) solve the expression

Example of how to solve this kind of log equation

1) rewrite both sides as single logs	logx + log23= log56
	log52x= log56
2) "cancel" logs	2x=6
3) Solve expression	x = 3

Problem 1) Solve the following equation :

log₃5 + log₃x = log₃15

Answer

Follow the steps for solving logarithmic equations with logs on both sides

1) rewrite both sides as single logs	log35x = log315
2) "cancel" logs	5x=15
3) Solve expression	x = 5

Problem 2) Solve the equation below:

log₃9 + log₃x = 4

Answer

Follow the steps for solving logarithmic equations with a log on one side

1 ) Rewrite log side as single logarithm	log39x = 4
2) Rewrite as exponential equation	34 = 9x
3) Solve exponential equation	81 =9x
	9 = x

Problem 3) Solve the following equation :

2log₃5 + log₃x = 3log₃5

Answer

Follow the steps on how to solve equations with logs on both sides

1) rewrite both sides as single logs	log352 +log3x = log353
	log325 +log3x = log3125
	log325x = log353
2) "cancel" logs	25x=125
3) Solve expression	x = 3

Problem 4) Solve the equation below:

2log₂4 + log₂x = 5

Answer

Follow the steps for solving logarithmic equations with a log on one side

1 ) Rewrite log side as single logarithm	2log24 + log2x = 5 log242 + log2x = 5 log216 + log2x = 5 log216x = 5
2) Rewrite as exponential equation	25 = 16x
3) Solve exponential equation	32 =16x
	2 = x

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Problem 5) Solve the following equation :

2log₃5 + log₃x = 3log₃5

Answer

Follow the steps on how to solve equations with logs on both sides

1) rewrite both sides as single logs	log352 +log3x = log353
	log325 +log3x = log3125
	log325x = log353
2) "cancel" logs	25x=125
3) Solve expression	x = 3

Problem 6) Solve the equation below:

2log₃7 - log₃2x = log₃2

Answer

Follow the steps on how to solve equations with logs on both sides

1) rewrite both sides as single logs	2log37 - log32x = log398
	log372 - log32x = log398
	log349 - log32x = log398
	log3(49/2x)= log398
2) "cancel" logs	(49/2x)= 98
3) Solve expression	49 = 196x
	49 = x

Problem 7) Solve :

2log₁₁5 + log₁₁x+ log₁₁2 = log₁₁150

Answer

You know the deal. Just follow the steps for solving logarithmic equations with logs on both sides

1) rewrite as single logs	2log115 + log11x+ log112 = log11150
	log1152 + log11x+ log112 = log11150
	log1125 + log112x = log11150
	log1150x = log11150
2) "cancel" logs	50x = 150
3) Solve expression	x = 3