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Mastering Exponential and Logarithmic Equations: Techniques for Solving

Learn strategies to solve exponential and logarithmic equations efficiently by rewriting them and applying inverse properties. Practice solving simple equations and checking for extraneous solutions in this comprehensive guide.

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Mastering Exponential and Logarithmic Equations: Techniques for Solving

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  1. 5.4 Exponential and Logarithmic Equations Essential Questions: How do we solve exponential and logarithmic equations?

  2. Solving Simple Equations Original Rewritten Solution Equation Equation

  3. Strategies for Solving Exponential and Logarithmic Equations • Rewrite the given equation in a form that allows the use of the One-to-One Properties of exponential or logarithmic functions. • Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions. • Rewrite a logarithmic equation in exponential form and apply the Inverse Property of exponential functions.

  4. Solution: a. Solving Exponential Equations • Solve each equation and approximate the result to three decimal places. a. b. b. Take natural log of each side. Use a calculator.

  5. Solution: Solving an Exponential Equation • Solve and approximate the result to three decimal places. Original equation. Subtract 5 from each side. Take natural log of each side. Inverse Property. Use a calculator.

  6. Solve and approximate the result to three decimal places.

  7. Solving a Logarithmic Equation a. Solve ln x = 2 b. Solve Solution: a. Exponentiate each side. b. Check this in the original equation.

  8. Solving a Logarithmic Equation • Solve and approximate the result to three decimal places. Solution: Exponentiate each side.

  9. Solving a Logarithmic Equation • Solve Solution: Exponentiate each side (base 5).

  10. Checking for Extraneous Solutions • Solve Solution: The solutions appear to be 5 and -4. However, when you check these in the original equation, only x = 5 works.

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