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.

### Properties for Solving Exponential and Logarithmic Equations

• b, x, and y are real numbers, b > 0, b ≠ 1
• If x = y, then bx = by
• If bx = by, then x = y
• If x = y and x > 0, y > 0, then logb(x) = logb(y)
• If x > 0, y > 0 and logb(x) = logb(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 logb(x) = logb(y), then x = y
• If logb(x) = k, then x = bk
Let's look at a few examples.
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 logb(x) = logb(y), then x = y
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 logb(x) = k, then x = bk
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. $$log_{9}(x - 8) + log_{9}(10)=log_{9}(80)$$

A
$$x=\frac{1}{2}$$
B
$$x=4$$
C
$$x=\frac{41}{4}$$
D
$$x=18$$
E
$$x=16$$

Example #2

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

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

A
$$x=-1$$
B
$$x=2$$
C
$$x=\frac{15}{2}$$
D
$$x=\frac{4}{5}$$
E
$$x=\frac{5}{3}$$