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Section 7.6 Notes

Section 7.6 Notes. Exponential equations are equations in which variable expressions occur as exponents . Property of Equality for Exponential Equations If b is a positive number other than 1, then b x = b y if and only if x = y. Example 1. Solve the equation. a.

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Section 7.6 Notes

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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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