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Unit 5: Modeling with Exponential & Logarithmic Functions

Unit 5: Modeling with Exponential & Logarithmic Functions. Ms. C. Taylor. Warm-Up. Identify the value of b in the following:. Graphing Exponential Equations. The graph will approach the axis but will never touch. Asymptote for the function will approach the x-axis.

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Unit 5: Modeling with Exponential & Logarithmic Functions

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  1. Unit 5: Modeling with Exponential & Logarithmic Functions Ms. C. Taylor

  2. Warm-Up • Identify the value of b in the following:

  3. Graphing Exponential Equations • The graph will approach the axis but will never touch. • Asymptote for the function will approach the x-axis. • Asymptote for the inverse function will approach the y-axis.

  4. Warm-Up • Rewrite using exponent rules

  5. Logarithms • Suppose b>0 and b≠1. For x>0, there is a number y such that if and only if

  6. LogarithmicExponential Form

  7. ExponentialLogarithmic Form

  8. Inverse Property of Exponents & Logarithms

  9. LogarithmicExponential Inequality • If • If

  10. Property of Equality for Logarithmic Functions • If b is a positive number other than 1, then if and only if • Example: If , then

  11. Property of Inequality for Logarithmic Functions • If , then if and only if, and if and only if • If , then

  12. Product Property of Logarithms • For all positive numbers m, n, and b, where b≠1,

  13. Example #1 • Expand the following logarithms:

  14. Example #2 • Use to approximate the value of • Use to approximate the value of

  15. Quotient Property of Logarithms • For all positive numbers m, n, and b, where ,

  16. Example #3 • Expand the following logarithms:

  17. Example #4 • Use and to approximate • Use and to approximate

  18. Power Property of Logarithms • For any real number p and positive numbers m and b, where ,

  19. Examples • Given , approximate the value of • Given , approximate the value of

  20. Warm-Up • Expand the following:

  21. Find Common Logarithms

  22. Change of Base Formula • For all positive numbers, a, b, and n, where and ,

  23. Examples • Express in terms of common logarithms. Then approximate its value to four decimal places. • Express in terms of common logarithms. Then approximate its value to four decimal places.

  24. Evaluate Natural Base Expressions

  25. Evaluate Natural Logarithmic Expressions

  26. Equivalent Expressions • If something has an e in it then that will become a ln. • If something has an ln in it then it will become e raised to a power.

  27. Warm-Up • Evaluate the following

  28. Warm-Up • Use the properties of logarithms to rewrite:

  29. Inverse Property of Base e & Natural Logarithms

  30. Evaluate Logarithmic Expressions

  31. Solve Logarithmic Equations

  32. Solve Equations with Logarithms on Both Sides • Solve

  33. Solve Equations using Properties of Logarithms

  34. Warm-Up

  35. Solve Exponential Equations using Logarithms

  36. Solve Base e Equations

  37. Solve Natural Log Equations & Inequalities

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