Lesson Objectives

- Demonstrate an understanding of exponents and logarithms
- Learn how to solve logarithmic equations with logarithms on each side
- Learn how to solve logarithmic equations with a logarithm equal to a number

## How to Solve Logarithmic Equations

Before we jump in and start solving logarithmic equations, let's look at some of the properties we will be using in this lesson.

Example 1: Solve each equation $$log_{5}(30)=log_{5}(3x + 9)$$ To solve this equation, we use the following property:

If x > 0, y > 0 and log

Since we have the same base on each log, we can set the arguments equal to each other: $$3x + 9=30$$ Subtract 9 away from each side of the equation: $$3x=21$$ Divide each side by 3: $$x=7$$ Example 2: Solve each equation $$9 - 3log_{8}(3x - 1)=6$$ For this scenario, we want to isolate the logarithm. Let's begin by subtracting 9 away from each side of the equation. $$-3log_{8}(3x - 1)=-3$$ Divide each side by -3: $$log_{8}(3x - 1)=1$$ To solve this equation, we use the following property:

If log

In other words, we write this in exponential form: $$3x - 1=8^1$$ $$3x - 1=8$$ Add 1 to each side of the equation: $$3x=9$$ Divide each side by 3: $$x=3$$

### Properties for Solving Exponential and Logarithmic Equations

- b, x, and y are real numbers, b > 0, b ≠ 1
- If x = y, then b
^{x}= b^{y} - If b
^{x}= b^{y}, then x = y - If x = y and x > 0, y > 0, then log
_{b}(x) = log_{b}(y) - If x > 0, y > 0 and log
_{b}(x) = log_{b}(y), then x = y

### Solving Logarithmic Equations

- Transform the equation so that a single logarithm appears on one side
- We can use the product rule or quotient rule for logarithms to accomplish this task

- Use one of the following rules to obtain a solution
- If x > 0, y > 0 and log
_{b}(x) = log_{b}(y), then x = y - If log
_{b}(x) = k, then x = b^{k}

- If x > 0, y > 0 and log

Example 1: Solve each equation $$log_{5}(30)=log_{5}(3x + 9)$$ To solve this equation, we use the following property:

If x > 0, y > 0 and log

_{b}(x) = log_{b}(y), then x = ySince we have the same base on each log, we can set the arguments equal to each other: $$3x + 9=30$$ Subtract 9 away from each side of the equation: $$3x=21$$ Divide each side by 3: $$x=7$$ Example 2: Solve each equation $$9 - 3log_{8}(3x - 1)=6$$ For this scenario, we want to isolate the logarithm. Let's begin by subtracting 9 away from each side of the equation. $$-3log_{8}(3x - 1)=-3$$ Divide each side by -3: $$log_{8}(3x - 1)=1$$ To solve this equation, we use the following property:

If log

_{b}(x) = k, then x = b^{k}In other words, we write this in exponential form: $$3x - 1=8^1$$ $$3x - 1=8$$ Add 1 to each side of the equation: $$3x=9$$ Divide each side by 3: $$x=3$$

#### Skills Check:

Example #1

Solve each equation. $$log_{9}(x - 8) + log_{9}(10)=log_{9}(80)$$

Please choose the best answer.

A

$$x=\frac{1}{2}$$

B

$$x=4$$

C

$$x=\frac{41}{4}$$

D

$$x=18$$

E

$$x=16$$

Example #2

Solve each equation. $$log_{6}(-4x) - log_{6}(2)=3$$

Please choose the best answer.

A

$$x=-\frac{3}{4}$$

B

$$x=36$$

C

$$x=-\frac{17}{2}$$

D

$$x=-\frac{15}{4}$$

E

$$x=-108$$

Example #3

Solve each equation. $$log_{5}(4x) - log_{5}(6)=1$$

Please choose the best answer.

A

$$x=-1$$

B

$$x=2$$

C

$$x=\frac{15}{2}$$

D

$$x=\frac{4}{5}$$

E

$$x=\frac{5}{3}$$

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