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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 a\bullet b^{x}=d
  • 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 log_b(b^{x})=x to simplify one side.
  • Step 4: If the base is NOT base 10 or e, use the change-of-base formula to rewrite the other side of the equation.
    • (log_c a=\frac{log_b a}{log_b c})
  • Step 5: Evaluate the equation to solve for the variable.
    • Round to the two decimal places.
Alg2B 9.6 COB Examples

 

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

Alg2B 9.6 Power Law Ex

  • Example with a = 8

Alg2B 9.6 Power Law Ex2

 

Examples

  • Ex 1. What is the first step in solving the equation below?
  • 5^{x}=21
    • Answer: Take the log of both sides: log\; 5^{x}=log\; 21
  • Ex 2. What is the solution to the equation below? Round your answer to two decimal places.
  • e^{x}=7.9
  • Setup: take the natural log (ln) of both sides, then use a calculator to solve.
    • Answer: x = 2.07
  • Ex 3. What is the solution to the equation below? Round your answer to two decimal places.
  • 5^{x}=9.2
  • Setup: Take the log base 5 of both sides, then use change-of-base and a calculator to solve.
    • log_5 5^{x}=log_5 9.2
    • x=\frac{log\; 9.2}{log\; 5}
      • Answer: x = 1.38
  • Ex 4. What is the solution to the equation below? Round your answer to two decimal places.
  • 5\bullet e^{x}=15.76
  • Setup: divide both sides by 5, then take the natural log of both sides to solve.
    • Answer: x = 1.15
  • Ex 5. What is the solution to the equation below? Round your answer to two decimal places.
  • 3\bullet 7^{x}=10.48
  • Setup: divide both sides by 3, then take the log base 7 of both sides. Set up change-of-base and solve.
    • You could also use the power law instead of change-of-base.
      • Answer: x = 0.64
  • Ex 6. What is the solution to the equation below? Round your answer to two decimal places.
  • 2^{3x}=73
  • Setup: Take the common log of both sides, then use the power law, then simplify and solve.
    • log\; 2^{3x}=log\; 73
    • 3x\bullet log\; 2=log\; 73
    • 3x\bullet \frac{log\; 2}{log\; 2}=\frac{log\; 73}{log\; 2}
    • 3x=\frac{1.86}{0.3}
    • 3x=6.19
      • Answer: x = 2.06
  • Ex 7. What is the solution to the equation below? Round your answer to two decimal places.
  • 3e^{2x}+12=42
  • Setup: Subtract 12 from both sides, then take the natural log of both sides, then use the power law, then simplify and solve.
    • \frac{3e^{2x}}{3}=\frac{30}{3}
    • e^{2x}=10
    • ln_e\; e^{2x}=ln_e\; 10
    • 2x=2.30
      • Answer: x = 1.15

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