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9.8 – Solving Exponential Equations
Key Terms
 No key terms for this section.
Review
Remember… 
 The inverse of an exponent is a logarithm, and vice versa.
 Whatever you do to ONE side of the equation, you MUST do to the OTHER side of the equation to keep the value unchanged.
 So, taking the logarithm of both sides of an equation does not change the value of the equation.

Notes
How to Solve Exponential Equations in the Form 
 Step 1: Isolate the exponential expression by dividing both sides by a.
 Sometimes a = 1, so you can skip this step.
 Step 2: Isolate the variable by taking the logarithm, base b, of both sides.
 This makes the log cancel out on one side (see Step 3).
 Step 3: Use the property to simplify one side.
 Step 4: If the base is NOT base 10 or e, use the changeofbase formula to rewrite the other side of the equation.
 Step 5: Evaluate the equation to solve for the variable.
 Round to the two decimal places.


Using the Power Property 
 You can solve an equation that has an expression with a base other than 10 or e without using the changeofbase formula.
 Divide both sides by a.
 If a = 1, skip to step 2
 Take the common logarithm of both sides.
 Use the power property to simplify the left side.
 Move the x to the front (power law)
 Isolate the variable.
 Divide by that log on both sides to isolate x
 Evaluate.
 Use a calculator to simplify the numerator and denominator, then solve



Examples
 Ex 1. What is the first step in solving the equation below?

 Answer: Take the log of both sides:

 Ex 2. What is the solution to the equation below? Round your answer to two decimal places.
 Setup: take the natural log (ln) of both sides, then use a calculator to solve.

 Ex 3. What is the solution to the equation below? Round your answer to two decimal places.
 Setup: Take the log base 5 of both sides, then use changeofbase and a calculator to solve.

 Ex 4. What is the solution to the equation below? Round your answer to two decimal places.
 Setup: divide both sides by 5, then take the natural log of both sides to solve.

 Ex 5. What is the solution to the equation below? Round your answer to two decimal places.
 Setup: divide both sides by 3, then take the log base 7 of both sides. Set up changeofbase and solve.
 You could also use the power law instead of changeofbase.

 Ex 6. What is the solution to the equation below? Round your answer to two decimal places.
 Setup: Take the common log of both sides, then use the power law, then simplify and solve.

 Ex 7. What is the solution to the equation below? Round your answer to two decimal places.
 Setup: Subtract 12 from both sides, then take the natural log of both sides, then use the power law, then simplify and solve.

Important!
Practice (Apex Study 9.8)
 Try practice problems on Pgs 7
 Mandatory: write and answer problems on Pgs 8
 1 Quiz
 Check out the rice / chess problem on pages 9 – 14
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