# 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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