### 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

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

rewrite both sides as single logs

$$ log_3 5x = log_3 15 $$

"cancel" logs

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

Solve expression

x = 3

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

Rewrite log side as single logarithm

$$ log_3 9x = 4$$

$$ 3^4 = 9x $$

81 = 9 x

9 = x

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

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 $

"cancel" logs

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

Solve expression

x = 3

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

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 $

2^{5} = 16x

32 =16x

2 = x

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

rewrite both sides as single logs

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

log_{3}25x = log_{3}5^{3}

"cancel" logs

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

Solve expression

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

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

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 $

"cancel" logs

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

Solve expression

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

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 $

2log_{11}5 + log_{11}x+ log_{11}2 = log_{11}150

"cancel" logs

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

Solve expression

x = 3