﻿ How To Solve Logarithmic Equations. Video Tutorial and Practice Problems

How To Solve Logarithmic Equations

Step By Step Video and practice Problems

How To Solve Logarithmic Equations Video

What is the general strategy for solving log equations?

Answer: 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

• A) has a logarithm on one side and a number on the other
• B) whether it has logarithms on both sides

Example 1 Logarithm on one side and a number on the other

General method for solving this type (log on one side), Rewrite the logarithm as exponential equation and solve. Let's look at a specific example:

$$log_4 x + log_4 8 = 3$$

Step 1 Rewrite log side as single logarithm

$$log_4 8x = 3$$

$$4^ 3 = 8x$$

64 = 8x
8 = x

Example 2 Logarithm on both sides

General method to solve this kind (logarithm on both sides),

Step 1 use the rules of logarithms to rewrite the left side and the right side of the equation to a single logarithm

Step 2 "cancel" the log

Step 3 solve the expression

Let's look at a specific ex $$log_5 x + log_2 3 = log_5 6$$

Step 1 rewrite both sides as single logs

$$log_5 x + log_5 2 = log_5 6 \\ log_5 2x = log_5 6$$

Step 2 "cancel" logs

$$\color{Red}{ \cancel {log_5}} 2x = \color{Red}{ \cancel {log_5}} 6 \\ 2x = 6$$

Step 2 Solve expression

x = 3

Practice Problems

Step 1

rewrite both sides as single logs

$$log_3 5x = log_3 15$$

Step 2

"cancel" logs

$$\color{Red}{ \cancel{log_3}} 5x = \color{Red}{ \cancel{log_3}} 15 \\ 5x=15$$

Step 3

Solve expression

x = 3

Step 1

Rewrite log side as single logarithm

$$log_3 9x = 4$$

Step 2

$$3^4 = 9x$$

Step 3

81 = 9 x
9 = x

Step 1

rewrite both sides as single logs<

$log_3 5^2 + log_3 x = log_3 5^3 \\ log_3 25 +log_3 x = log_3 125 \\ log_3 25x = log_3 125$

Step 2

"cancel" logs

$\color{Red}{ \cancel{log_3}} 25x = \color{Red}{ \cancel{log_3}} 125 \\ 25x = 125$

Step 3

Solve expression

x = 3

Step 1

Rewrite log side as single logarithm

$2 log_2 4 + log_2 x = 5 \\ log_2 4^2 = log_2 x = 5 \\ log_2 16 + log_2 x = 5 \\ log_2 16x = 5$

Step 2

25 = 16x

Step 3

32 =16x
2 = x

Step 1

rewrite both sides as single logs

$log_3 5^2 + log_3 x + log_3 5^3 \\ log_3 25x + log_3 125$

log325x = log353

Step 2

"cancel" logs

$\color{Red}{ \cancel{log_3}} 25x + \color{Red}{ \cancel{log_3}} 125 \\ 25x=125$

Step 3

Solve expression

$\frac{25x}{25} = \frac{125}{25} \\$

Step 1

rewrite both sides as single logs

$2 log_3 7 - log_3 2x = log_3 98 \\ log_3 7^2 - log_3 2x = log_3 98 \\ log_3 49 - log_3 2x = log_3 98 \\ log_3 \frac{49}{2x} = log_3 98$

Step 2

"cancel" logs

$\color{Red}{ \cancel{log_3}} \frac{49}{2x} = \color{Red}{ \cancel{log_3}} 98 \\ \frac{49}{2x} = 98$

Step 3

Solve expression

$49 = 196x \\ \frac{49}{196} = x \\ x = 49$

Step 1

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

rewrite as single logs

$2 log_11 5 + log_11 x + log_11 2 = log_11 150 \\ log_11 5^2 + log_11 2x = log_11 150 \\ log_11 25 + log_11 2x = log_11 150 \\ log_11 50x= log_11 150$

2log115 + log11x+ log112 = log11150

Step 2

"cancel" logs

$\color{Red}{ \cancel{log_1}} 50x = \color{Red}{ \cancel{log_11}} 150 \\ 50x = 150$

Step 3

Solve expression

x = 3

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