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Solving Exponential and Logarithmic Equations: Properties and Examples

This section explains how to solve exponential equations, which feature variable expressions as exponents. It outlines the property of equality for exponential equations and provides detailed steps with examples. The process of solving logarithmic equations is also covered, including the relevant property of equality. Examples demonstrate how to transform and solve equations involving logarithms and highlight the importance of checking for extraneous solutions. The use of logarithls and calculators is emphasized for various forms of these equations.

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Solving Exponential and Logarithmic Equations: Properties and Examples

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  1. Section 7.6 Notes

  2. Exponential equationsare equations in which variable expressions occur as exponents. • Property of Equality for Exponential Equations • If b is a positive number other than 1, then bx = byif and only if x = y.

  3. Example 1 • Solve the equation.

  4. a. • Step 1: Get the bases to be the same number. • Step 2: Use the property of equality for exponential equations to solve for x.

  5. Example 2 • Solve the equation. Round to the nearest thousandth.

  6. a. 2x = 5 • Step 1: Take the log2 of each side. • Step 2: Use a calculator to solve.

  7. b. 79x = 15

  8. c. 4e-0.3x – 7 = 13 • Step 1: Get the number raised to a power by itself. • Step 2: Solve for the variable.

  9. Logarithmic equationsare equations that involve logarithms of variable expressions. • Property of Equality for Logarithmic Equations • If b, x, and y are positive numbers with b ≠ 1, then logb x = logb y if and only if x = y.

  10. Example 3 • Solve.

  11. b. ln (7x – 4) = ln (2x + 11)

  12. The property of equality for exponential equations implies that if you are given an equation • x = y, then your can exponentiate each side to obtain an equation in the form bx = by. This technique is useful for solving some logarithmic equations.

  13. Example 4 • Solve.

  14. a. log7 (3x – 2) = 2

  15. b. log2 (x – 6) = 5

  16. Because the domain of a logarithmic function generally does not include all real numbers, be sure to check for extraneous solutions of logarithmic equations.

  17. Example 5 • Solve.

  18. a. log6 3x + log6 (x – 4) = 2 • Step 1: Use a property of logarithms to make the left side into a single logarithm. • Step 2: Solve the new equation.

  19. c. log4 (x + 12) + log4x = 3

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