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Section5.5 Solving Exponential Equations and Logarithmic Equa- - PowerPoint PPT Presentation

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Section5.5 Solving Exponential Equations and Logarithmic Equa- tions ExponentialEquations Definition An exponential equation is one where the variable appears in an exponent. Definition An exponential equation is one where the variable appears

  1. Section5.5 Solving Exponential Equations and Logarithmic Equa- tions

  2. ExponentialEquations

  3. Definition An exponential equation is one where the variable appears in an exponent.

  4. Definition An exponential equation is one where the variable appears in an exponent. There are a few different categories that these equations can fall into.

  5. The equation simplifies to a X = a Y Set X = Y and finish solving.

  6. The equation simplifies to a X = a Y Set X = Y and finish solving. For example: 3 x 2 = 9 2 x − 2 5 x − 2 = 23 5 x = 25 3 x 2 = (3 2 ) 2 x − 2 5 x = 5 2 3 x 2 = 3 4 x − 4 x = 2 x 2 = 4 x − 4 x 2 − 4 x + 4 = 0 ( x − 2) 2 = 0 x = 2

  7. The equation simplifies to a X = b Y Apply your favorite type of log on both sides.

  8. The equation simplifies to a X = b Y Apply your favorite type of log on both sides. Bring down the exponents using the third Law of Logarithms

  9. The equation simplifies to a X = b Y Apply your favorite type of log on both sides. Bring down the exponents using the third Law of Logarithms Finish solving.

  10. The equation simplifies to a X = b Y Apply your favorite type of log on both sides. Bring down the exponents using the third Law of Logarithms Finish solving. For example: 4 x +1 = 7 1 − 2 x ln 4 x +1 = ln 7 1 − 2 x ( x + 1) ln 4 = (1 − 2 x ) ln 7 x ln 4 + ln 4 = ln 7 − 2 x ln 7 x ln 4 + 2 x ln 7 = ln 7 − ln 4 x (ln 4 + 2 ln 7) = ln 7 − ln 4 x = ln 7 − ln 4 ln 4 + 2 ln 7

  11. Quadratic-Type Exponential Equations Use a substitution to make the equation a quadratic.

  12. Quadratic-Type Exponential Equations Use a substitution to make the equation a quadratic. Solve for the new variable.

  13. Quadratic-Type Exponential Equations Use a substitution to make the equation a quadratic. Solve for the new variable. Go back to the original variable and finish solving.

  14. Quadratic-Type Exponential Equations Use a substitution to make the equation a quadratic. Solve for the new variable. Go back to the original variable and finish solving. For example: e 2 x − 4 e x − 5 = 0 ( e x ) 2 − 4 e x − 5 = 0 Make the substitution y = e x y 2 − 4 y − 5 = 0 ( y − 5)( y + 1) = 0 y = 5 or y = − 1 e x = 5 or e x = − 1 x = ln 5 or x = ln( − 1)

  15. Examples 1. 4 3 x − 5 = 16

  16. Examples 1. 4 3 x − 5 = 16 x = 7 3

  17. Examples 1. 4 3 x − 5 = 16 x = 7 3 2. 27 = 3 5 x · 9 x 2

  18. Examples 1. 4 3 x − 5 = 16 x = 7 3 2. 27 = 3 5 x · 9 x 2 x = 1 2 or x = − 3

  19. Examples 1. 4 3 x − 5 = 16 x = 7 3 2. 27 = 3 5 x · 9 x 2 x = 1 2 or x = − 3 3. e 4 t = 1000

  20. Examples 1. 4 3 x − 5 = 16 x = 7 3 2. 27 = 3 5 x · 9 x 2 x = 1 2 or x = − 3 3. e 4 t = 1000 t = ln 1000 4

  21. Examples 1. 4 3 x − 5 = 16 x = 7 3 2. 27 = 3 5 x · 9 x 2 x = 1 2 or x = − 3 3. e 4 t = 1000 t = ln 1000 4 4. 5 x +2 = 4 1 − x

  22. Examples 1. 4 3 x − 5 = 16 x = 7 3 2. 27 = 3 5 x · 9 x 2 x = 1 2 or x = − 3 3. e 4 t = 1000 t = ln 1000 4 4. 5 x +2 = 4 1 − x ln 4 x = ln 4 − ln 5 ln 5+ln 4 = 5 ln 20

  23. Examples 1. 4 3 x − 5 = 16 x = 7 3 2. 27 = 3 5 x · 9 x 2 x = 1 2 or x = − 3 3. e 4 t = 1000 t = ln 1000 4 4. 5 x +2 = 4 1 − x ln 4 x = ln 4 − ln 5 ln 5+ln 4 = 5 ln 20 5. e x + e − x = 4

  24. Examples 1. 4 3 x − 5 = 16 x = 7 3 2. 27 = 3 5 x · 9 x 2 x = 1 2 or x = − 3 3. e 4 t = 1000 t = ln 1000 4 4. 5 x +2 = 4 1 − x ln 4 x = ln 4 − ln 5 ln 5+ln 4 = 5 ln 20 5. e x + e − x = 4 √ √ x = ln(2 − 3) ≈ − 1 . 317 or x = ln(2 + 3) ≈ 1 . 317

  25. LogarithmicEquations

  26. Definition A logarithmic equation is one where the variable appears inside a logarithm.

  27. Definition A logarithmic equation is one where the variable appears inside a logarithm. For all these questions, you must check your answers against the domain of the original equation. There can be fake solutions!

  28. Definition A logarithmic equation is one where the variable appears inside a logarithm. For all these questions, you must check your answers against the domain of the original equation. There can be fake solutions! There are a few possible types that we will be solving.

  29. A logarithm is Raised to a Power Solve using a substitution.

  30. A logarithm is Raised to a Power Solve using a substitution. For example: (log 3 x ) 2 − log 3 x 2 = 3 Domain restrictions: x > 0 and x 2 > 0 (log 3 x ) 2 − 2 log 3 x = 3 Make the substitution y = log 3 x y 2 − 2 y = 3 y 2 − 2 y − 3 = 0 ( y − 3)( y + 1) = 0 y = 3 or y = − 1 log 3 x = 3 or log 3 x = − 1 x = 3 3 or x = 3 − 1 x = 27 or x = 1 3

  31. A logarithm is Raised to a Power Solve using a substitution. For example: (log 3 x ) 2 − log 3 x 2 = 3 Domain restrictions: x > 0 and x 2 > 0 (log 3 x ) 2 − 2 log 3 x = 3 Check x = 27: Make the substitution y = log 3 x y 2 − 2 y = 3 y 2 − 2 y − 3 = 0 ( y − 3)( y + 1) = 0 y = 3 or y = − 1 log 3 x = 3 or log 3 x = − 1 x = 3 3 or x = 3 − 1 x = 27 or x = 1 3

  32. A logarithm is Raised to a Power Solve using a substitution. For example: (log 3 x ) 2 − log 3 x 2 = 3 Domain restrictions: x > 0 and x 2 > 0 (log 3 x ) 2 − 2 log 3 x = 3 Check x = 27: Make the substitution y = log 3 x y 2 − 2 y = 3 27 > 0 y 2 − 2 y − 3 = 0 ( y − 3)( y + 1) = 0 y = 3 or y = − 1 log 3 x = 3 or log 3 x = − 1 x = 3 3 or x = 3 − 1 x = 27 or x = 1 3

  33. A logarithm is Raised to a Power Solve using a substitution. For example: (log 3 x ) 2 − log 3 x 2 = 3 Domain restrictions: x > 0 and x 2 > 0 (log 3 x ) 2 − 2 log 3 x = 3 Check x = 27: Make the substitution y = log 3 x y 2 − 2 y = 3 27 > 0 27 2 > 0 y 2 − 2 y − 3 = 0 ( y − 3)( y + 1) = 0 y = 3 or y = − 1 log 3 x = 3 or log 3 x = − 1 x = 3 3 or x = 3 − 1 x = 27 or x = 1 3

  34. A logarithm is Raised to a Power Solve using a substitution. For example: (log 3 x ) 2 − log 3 x 2 = 3 Domain restrictions: x > 0 and x 2 > 0 (log 3 x ) 2 − 2 log 3 x = 3 Check x = 27: Make the substitution y = log 3 x y 2 − 2 y = 3 27 > 0 27 2 > 0 y 2 − 2 y − 3 = 0 Check x = 1 3 : ( y − 3)( y + 1) = 0 y = 3 or y = − 1 log 3 x = 3 or log 3 x = − 1 x = 3 3 or x = 3 − 1 x = 27 or x = 1 3

  35. A logarithm is Raised to a Power Solve using a substitution. For example: (log 3 x ) 2 − log 3 x 2 = 3 Domain restrictions: x > 0 and x 2 > 0 (log 3 x ) 2 − 2 log 3 x = 3 Check x = 27: Make the substitution y = log 3 x y 2 − 2 y = 3 27 > 0 27 2 > 0 y 2 − 2 y − 3 = 0 Check x = 1 3 : ( y − 3)( y + 1) = 0 1 3 > 0 y = 3 or y = − 1 log 3 x = 3 or log 3 x = − 1 x = 3 3 or x = 3 − 1 x = 27 or x = 1 3

  36. A logarithm is Raised to a Power Solve using a substitution. For example: (log 3 x ) 2 − log 3 x 2 = 3 Domain restrictions: x > 0 and x 2 > 0 (log 3 x ) 2 − 2 log 3 x = 3 Check x = 27: Make the substitution y = log 3 x y 2 − 2 y = 3 27 > 0 27 2 > 0 y 2 − 2 y − 3 = 0 Check x = 1 3 : ( y − 3)( y + 1) = 0 1 3 > 0 y = 3 or y = − 1 � 1 � 2 > 0 log 3 x = 3 or log 3 x = − 1 3 x = 3 3 or x = 3 − 1 x = 27 or x = 1 3

  37. No Logarithms Raised to a Power 1. If there are any constant terms, move all those to one side and the log terms to the other side.

  38. No Logarithms Raised to a Power 1. If there are any constant terms, move all those to one side and the log terms to the other side. 2. Combine all the log terms on each side into a single logarithm using the Laws of Logarithms.

  39. No Logarithms Raised to a Power 1. If there are any constant terms, move all those to one side and the log terms to the other side. 2. Combine all the log terms on each side into a single logarithm using the Laws of Logarithms. 3. At this point, it should be in one of two forms:

  40. No Logarithms Raised to a Power 1. If there are any constant terms, move all those to one side and the log terms to the other side. 2. Combine all the log terms on each side into a single logarithm using the Laws of Logarithms. 3. At this point, it should be in one of two forms: log a X = Y . Rewrite as a Y = X and finish solving.

  41. No Logarithms Raised to a Power 1. If there are any constant terms, move all those to one side and the log terms to the other side. 2. Combine all the log terms on each side into a single logarithm using the Laws of Logarithms. 3. At this point, it should be in one of two forms: log a X = Y . Rewrite as a Y = X and finish solving. log a X = log a Y . Set X = Y and finish solving.

  42. No Logarithms Raised to a Power (continued) For example: Domain restrictions: ln( x + 8) + ln( x − 1) = 2 ln x x + 8 > 0, x − 1 > 0, x > 0 ln(( x + 8)( x − 1)) = ln x 2 ln( x 2 + 7 x − 8) = ln x 2 x 2 + 7 x − 8 = x 2 7 x − 8 = 0 7 x = 8 x = 8 7

  43. No Logarithms Raised to a Power (continued) For example: Domain restrictions: ln( x + 8) + ln( x − 1) = 2 ln x x + 8 > 0, x − 1 > 0, x > 0 ln(( x + 8)( x − 1)) = ln x 2 Check x = 8 ln( x 2 + 7 x − 8) = ln x 2 7 : x 2 + 7 x − 8 = x 2 7 x − 8 = 0 7 x = 8 x = 8 7

  44. No Logarithms Raised to a Power (continued) For example: Domain restrictions: ln( x + 8) + ln( x − 1) = 2 ln x x + 8 > 0, x − 1 > 0, x > 0 ln(( x + 8)( x − 1)) = ln x 2 Check x = 8 ln( x 2 + 7 x − 8) = ln x 2 7 : x 2 + 7 x − 8 = x 2 7 + 8 = 64 8 7 > 0 7 x − 8 = 0 7 x = 8 x = 8 7

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