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Section 4.3

Section 4.3. Logarithmic Functions and Graphs. Flashback. Consider the graph of the exponential function y = f ( x ) = 3 x . Is f(x) one-to-one? Does f(x) have an inverse that is a function? Find the inverse. Inverse of y = 3 x. f ( x ) = 3 x y = 3 x x = 3 y.

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Section 4.3

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  1. Section 4.3 Logarithmic Functions and Graphs

  2. Flashback Consider the graph of the exponential function y = f(x) = 3x. • Is f(x) one-to-one? • Does f(x) have an inverse that is a function? • Find the inverse.

  3. Inverse of y = 3x f (x) = 3x y = 3x x = 3y

  4. x = 3y Now, solve for y. y= the power to which 3 must be raised in order to obtain x.

  5. x = 3y Solve for y. y= the power to which 3 must be raised in order to obtain x. Symbolically, y = log 3 x “The logarithm, base 3, of x.”

  6. Logarithm For all positive numbers a, where a 1, Logax is an exponentto which the base a must be raised to give x.

  7. Logarithmic Form Exponential Form Argument (always positive) All a log is . . . is an exponent!

  8. Logarithmic Functions • Logarithmic functions are inverses of exponential functions. Graph: x = 3y ory = log 3 x 1. Choose values for y. 2. Compute values for x. 3. Plot the points and connect them with a smooth curve. * Note that the curve does not touch or cross the y-axis.

  9. x = 3y y (x, y) 1 0 (1, 0) 3 1 (3, 1) 9 2 (9, 2) 1/3 1 (1/3, 1) 1/9 2 (1/9, 2) 1/27 3 (1/27, 3) Logarithmic Functions continued Graph: x = 3y y = log 3x

  10. Side-by-Side Comparison f (x) = 3x f (x) = log3x

  11. Comparing Exponential and Logarithmic Functions

  12. Logarithmic Functions • Remember: Logarithmic functions are inverses of exponential functions.

  13. Asymptotes • Recall that the horizontal asymptote of the exponential function y = axis the x-axis. • The vertical asymptote of a logarithmic function is the y-axis.

  14. Logarithms • Convert each of the following to a logarithmic equation. a) 25 = 5xb) ew = 30

  15. The logarithm is the exponent. The base remains the same. Example • Convert each of the following to an exponential equation. • a) log7 343 = 3 log7343 = 37 3 = 343 b) logb R = 12

  16. Finding Certain Logarithms • Find each of the following. a) log2 16 b) log10 1000 c) log16 4 d) log10 0.001

  17. Common Logarithm Logarithms, base 10, are called common logarithms. • Log button on your calculator • is the common log *

  18. Function Value Readout Rounded log 723,456 5.859412123 5.8594 log 0.0000245 4.610833916 4.6108 log (4) ERR: nonreal ans Does not exist Example • Find each of the following common logarithms on a calculator. Round to four decimal places. a) log 723,456 b) log 0.0000245 c) log (4)

  19. Natural Logarithms • Logarithms, base e, are called natural logarithms. • The abbreviation “ln” is generally used for natural logarithms. • Thus, ln x means loge x. * ln button on your calculator is the natural log *

  20. Function Value Readout Rounded ln 723,456 13.49179501 13.4918 ln 0.0000245 10.61683744 10.6168 ln (4) ERR: nonreal ans Does not exist Example • Find each of the following natural logarithms on a calculator. Round to four decimal places. a) ln 723,456 b) ln 0.0000245 c) ln (4)

  21. Changing Logarithmic Bases • The Change-of-Base Formula For any logarithmic bases a and b, and any positive number M, Use change of base formula when you have a logarithm that is not base 10 or e.

  22. Example Find log6 8 using common logarithms. Solution: First, we let a = 10, b = 6, and M = 8. Then we substitute into the change-of-base formula:

  23. Example We can also use base e for a conversion. Find log6 8 using natural logarithms. Solution: Substituting e for a, 6 for b and 8 for M, we have

  24. Properties of Logarithms

  25. Graph: y = f(x) = log6 x. Select y. Compute x. x, or6 y y 1 0 6 1 36 2 216 3 1/6 1 1/36 2 Graphs of Logarithmic Functions

  26. Example • Graph each of the following. • Describe how each graph can be obtained from the graph of y = ln x. • Give the domain and the vertical asymptote of each function. • a) f(x) = ln (x 2) • b) f(x) = 2  ¼ ln x • c) f(x) = |ln (x + 1)|

  27. The graph is a shift 2 units right. The domain is the set of all real numbers greater than 2. The line x = 2 is the vertical asymptote. x f(x) 2.25 1.386 2.5 0.693 3 0 4 0.693 5 1.099 Graph f(x) = ln (x 2)

  28. The graph is a vertical shrinking, followed by a reflection across the x-axis, and then a translation up 2 units. The domain is the set of all positive real numbers. The y-axis is the vertical asymptote. x f(x) 0.1 2.576 0.5 2.173 1 2 3 1.725 5 1.598 Graph f(x) = 2  ¼ ln x

  29. The graph is a translation 1 unit to the left. Then the absolute value has the effect of reflecting negative outputs across the x-axis. The domain is the set of all real numbers greater than 1. The line x = 1 is the vertical asymptote. x f(x) 0.5 0.693 0 0 1 0.693 3 1.386 6 1.946 Graph f(x) = |ln (x + 1)|

  30. Application: Walking Speed • In a study by psychologists Bornstein and Bornstein, it was found that the average walking speed w, in feet per second, of a person living in a city of population P, in thousands, is given by the function w(P) = 0.37 ln P + 0.05.

  31. Application: Walking Speed continued The population of Philadelphia, Pennsylvania, is 1,517,600. Find the average walking speed of people living in Philadelphia. Since 1,517,600 = 1517.6 thousand, we substitute 1517.6 for P, since P is in thousands: w(1517.6) = 0.37 ln 1517.6+0.05  2.8 ft/sec. The average walking speed of people living in Philadelphia is about 2.8 ft/sec.

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