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Understanding Exponential and Logarithmic Equations: One-to-One Properties and Inverses

This guide explains the one-to-one properties of exponential and logarithmic equations, detailing how to solve for variables using natural logarithms and understanding inverse relationships. Examples include solving equations of the forms ( e^x = 8 ), ( 3^x = 64 ), and ( e^{2.724x} = 29 ). Learn how the inverse property states that if ( ln(a) = ln(b) ), then ( a = b ). It also covers practical applications of logarithmic functions and the quadratic formula for solving more complex equations. Ideal for homework support.

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Understanding Exponential and Logarithmic Equations: One-to-One Properties and Inverses

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  1. 3.4 Exponential and Logarithmic Equations One-to-One Properties Inverse Property

  2. One-to-One Properties If x 6 = x y , then 6 = y If ln a = ln b, then a = b

  3. Inverse Property Given e x = 8; solve for x Take the natural log of each side. ln e x = ln 8 Pull the exponent in front x ( ln e) = ln 8 (since ln e = 1) x = ln 8

  4. Solve for x 3 x = 64 take the natural log of both sides ln 3 x = ln 64 x( ln 3) = ln 64 x = ln 64 = 3.7855.. ln 3

  5. Solve for x e x – 8 = 70

  6. Solve for x e x – 8 = 70 e x = 78 ln e x = ln 78 x = ln 78 x = 4.3567..

  7. Solve for a ( ¼ ) a = 64

  8. Solve for K Log 5 K = - 3

  9. Solve for x 2 x – 3 = 32

  10. Solve for x

  11. Solve for x e 2.724x = 29

  12. Solve for a ln a + ln ( a + 3) = 1 Will need the quadratic formula

  13. Solve for x one more time e 2x – e x – 12 = 0 factor

  14. Solve for x one more time e 2x – e x – 12 = 0 factor (e x – 4)(e x + 3 ) = 0 So e x – 4 = 0 or e x + 3 = 0 e x = 4 e x = - 3 x = ln 4 x = ln -3

  15. Homework Page 233 – 235 # 5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 105, 115, 125, 135

  16. Homework Page 233 – 235 # 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140

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