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Section 4.6 Logarithmic Functions

Section 4.6 Logarithmic Functions. Objectives: 1. To convert between exponential and logarithmic form. 2. To evaluate logarithms of numbers. 3. To graph logarithmic functions and give their domains and ranges.

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Section 4.6 Logarithmic Functions

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  1. Section 4.6 Logarithmic Functions

  2. Objectives: 1. To convert between exponential and logarithmic form. 2. To evaluate logarithms of numbers. 3. To graph logarithmic functions and give their domains and ranges.

  3. The inverse of the exponential function is the logarithmic function.

  4. f(x) = 2x y = x f(x) = log2x

  5. Definition Logarithmic function f(x) = logax, where a > 0, a  1, and x > 0. It is the inverse of the exponential function f(x) = ax.

  6. The rule y = logax is equivalent to ay = x; therefore, a is the base of the logarithm and y, the logarithm, is an exponent.

  7. The logarithmic expression can be written in either of two forms. Log Form Exponential Form logax = y ↔ay = x

  8. EXAMPLE 1Write log4 64 = y in exponential form. log4 64 = y 4y = 64

  9. EXAMPLE 2Write 3-5 = in logarithmic form. 3-5 = 1 243 1 243 1 243 log3 = -5

  10. EXAMPLE 3Prove loga 1 = 0  a  0 with a  1. a0 = 1 loga 1 = 0

  11. A logarithm in base ten is called a common logarithm. The base of a common log is not usually written. Common logs can be found using the log key on your calculator.

  12. The second special type of logarithm is the natural logarithm whose base is e. Recall that e is an irrational number that is approximately 2.71828. The special notation for a natural logarithm is ln.

  13. n æ 1 ö ç ÷ e = lim 1 + n è ø n 1 1 1 1 ¥ å + + + e = = 1 + 1 + ∙∙∙ n! 2 6 24 n=0 e, also called the Euler number, is defined as or

  14. Homework pp. 202-203

  15. ►A. Exercises Change the following logarithms to exponential form. 1. log2x = y

  16. 1 16 ►A. Exercises Change the following logarithms to exponential form. 3. log4 = -2

  17. ►A. Exercises Change the following logarithms to exponential form. 5. log 1000 = 3

  18. ►A. Exercises Change the following logarithms to exponential form. 7. ln 1 = 0

  19. ►A. Exercises Explain why the following are true. 9. loga a = 1

  20. ►A. Exercises Change the following exponential expressions to log form. 11. 53 = 125

  21. ►A. Exercises Change the following exponential expressions to log form. 13. 82 = 64

  22. ►A. Exercises Change the following exponential expressions to log form. 15. 73 = 343

  23. ►B. Exercises Graph. Give the domain and range of each. 17. y = log4x

  24. ►B. Exercises Graph. Give the domain and range of each. 17. y = log4x

  25. ►B. Exercises Evaluate. 21. log5 1,953,125 5y = 1,953,125 5y = 59 y = 9

  26. ►B. Exercises Evaluate. 23. log2 1 8 2y = 8-1 2y = (23)-1 2y = 2-3 y = -3

  27. ►B. Exercises Evaluate. 25. log6 1 1296

  28. ■Cumulative Review Use interval notation to show the intervals of continuity of the following functions. 29. f(x) = 3x2 – 5 x

  29. ■Cumulative Review Use interval notation to show the intervals of continuity of the following functions. 30. g(x) = 3x2 – 7x + 9

  30. ■Cumulative Review Use interval notation to show the intervals of continuity of the following functions. 31. h(x) = x – 3

  31. ■Cumulative Review 32. Name three characteristics of a graph that cause a function to be discontinuous. Name an example of each.

  32. ■Cumulative Review 33. Certain functions, such as the absolute value function, do not graph as a smooth curve, but have sharp turns in their graphs. Is such a function discontinuous?

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