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4 th EDITION

College Algebra & Trigonometry and Precalculus. 4 th EDITION. 4.3. Logarithmic Functions. Logarithms Logarithmic Equations Logarithmic Functions Properties of Logarithms. Logarithms.

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4 th EDITION

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  1. College Algebra & Trigonometry and Precalculus 4th EDITION

  2. 4.3 Logarithmic Functions Logarithms Logarithmic Equations Logarithmic Functions Properties of Logarithms

  3. Logarithms The previous section dealt with exponential functions of the form y = ax for all positive values of a, where a ≠ 1. The horizontal line test shows that exponential functions are one-to-one, and thus have inverse functions.

  4. Logarithms The equation defining the inverse of a function is found by interchanging x and y in the equation that defines the function. Starting with y = ax and interchanging x and y yields

  5. Logarithms Here y is the exponent to which a must be raised in order to obtain x. We call this exponent a logarithm, symbolized by “log.” The expression loga x represents the logarithm in this discussion. The number a is called the base of the logarithm, and x is called the argument of the expression. It is read “logarithm with base a of x,” or“logarithm of x with base a.”

  6. Logarithm For all real numbers y and all positive numbers a and x, where a ≠ 1, if and only if A logarithm is an exponent. The expression logax represents the exponent to which the base “a” must be raised in order to obtain x.

  7. Logarithms Exponent Logarithmic form: y = logax Base Exponent Exponential form: ay = x Base

  8. Logarithms

  9. SOLVING LOGARITHMIC EQUATIONS Example 1 Solve a. Solution Write in exponential form. Take cube roots

  10. SOLVING LOGARITHMIC EQUATIONS Example 1 Original equation Check: ? Let x = ⅔. ? Write in exponential form True The solution set is

  11. SOLVING LOGARITHMIC EQUATIONS Example 1 Solve b. Solution Write in exponential form. The solution set is {32}.

  12. SOLVING LOGARITHMIC EQUATIONS Example 1 Solve c. Solution Write in exponential form. Write with the same base. Power rule for exponents. The solution set is Set exponents equal. Divide by 2.

  13. Logarithmic Function If a > 0, a ≠ 1, and x > 0, then defines the logarithmic function with base a.

  14. Logarithmic Function Exponential and logarithmic functions are inverses of each other. The graph of y = 2x is shown in red. The graph of its inverse is found by reflecting the graph across the line y = x.

  15. Logarithmic Function The graph of the inverse function, defined by y = log2x, shown in blue, has the y-axis as a vertical asymptote.

  16. Logarithmic Function Since the domain of an exponential function is the set of all real numbers, the range of a logarithmic function also will be the set of all real numbers. In the same way, both the range of an exponential function and the domain of a logarithmic function are the set of all positive real numbers, so logarithms can be found for positive numbers only.

  17. Domain: (0, ) Range: (– , ) LOGARITHMIC FUNCTION For (x) = log2x: • (x) = logax, a > 1, is increasing and continuous on its entire domain, (0, ) .

  18. Domain: (0, ) Range: (– , ) LOGARITHMIC FUNCTION For (x) = log2x: • The y-axis is a vertical asymptote as x  0 from the right.

  19. Domain: (0, ) Range: (– , ) LOGARITHMIC FUNCTION For (x) = log2x: • The graph passes through the points

  20. Domain: (0, ) Range: (– , ) LOGARITHMIC FUNCTION For (x) = log1/2x: • (x) = logax, 0 < a < 1, is decreasing and continuous on its entire domain, (0, ) .

  21. Domain: (0, ) Range: (– , ) LOGARITHMIC FUNCTION For (x) = log1/2x: • The y-axis is a vertical asymptote as x  0 from the right.

  22. Domain: (0, ) Range: (– , ) LOGARITHMIC FUNCTION For (x) = log1/2x: • The graph passes through the points

  23. Characteristics of the Graph of 1. The points are on the graph. 2. If a > 1, then is an increasing function; if 0 < a < 1, then  is a decreasing function. 3. The y-axis is a vertical asymptote. 4. The domain is (0,), and the range is (–, ).

  24. GRAPHING LOGARITHMIC FUNCTIONS Example 2 Graph the function. a. Solution First graph y = (½)x which defines the inverse function of , by plotting points. The graph of (x) = log1/2x is the reflection of the graph y = (½)x across the line y = x. The ordered pairs for are found by interchanging the x- and y-values in the ordered pairs for y = (½)x .

  25. GRAPHING LOGARITHMIC FUNCTIONS Example 2 Graph the function. a. Solution

  26. GRAPHING LOGARITHMIC FUNCTIONS Example 2 Graph the function. b. Solution Another way to graph a logarithmic function is to write (x) = y = log3xin exponential form as x = 3y.

  27. GRAPHING LOGARITHMIC FUNCTIONS Example 2 Graph the function. a. Solution

  28. CautionIf you write a logarithmic function in exponential form, choosing y-values to calculate x-values, be careful to write the values in the ordered pairs in the correct order.

  29. GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Example 3 Graph each function. Give the domain and range. a. Solution The graph of (x) = log2 (x – 1) is the graph of (x) = log2 x translated 1 unit to the right. The vertical asymptote is x = 1. The domain of this function is (1, ) since logarithms can be found only for positive numbers. To find some ordered pairs to plot, use the equivalent exponential form of the equation y = log2 (x – 1).

  30. GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Example 3 Graph each function. Give the domain and range. a. Solution Write in exponential form. Add 1.

  31. GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Example 3 Graph each function. Give the domain and range. a. Solution We choose values for y and then calculate each of the corresponding x-values. The range is (–, ).

  32. GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Example 3 Graph each function. Give the domain and range. b. Solution The function defined by (x) = (log3x) – 1has the same graph as g(x) = log3xtranslated 1 unit down. We find ordered pairs to plot by writing y= (log3x) – 1 in exponential form.

  33. GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Example 3 Graph each function. Give the domain and range. b. Solution Add 1. Write in exponential form.

  34. GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Example 3 Graph each function. Give the domain and range. b. Solution Again, choose y-values and calculate the corresponding x-values. The domain is (0, ) and the range is (–, ).

  35. Properties of Logarithms Since a logarithmic statement can be written as an exponential statement, it is not surprising that the properties of logarithms are based on the properties of exponents. The properties of logarithms allow us to change the form of logarithmic statements so that products can be converted to sums, quotients can be converted to differences, and powers can be converted to products.

  36. Properties of Logarithms For x > 0, y > 0, a > 0, a ≠1, and any real number r: Description The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. Property Product Property

  37. Properties of Logarithms For x > 0, y > 0, a > 0, a ≠1, and any real number r: Description The logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numbers. Property Quotient Property

  38. Properties of Logarithms For x > 0, y > 0, a > 0, a ≠1, and any real number r: Description The logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number. Property Power Property

  39. Properties of Logarithms Two additional properties of logarithms follow directly from the definition of logax since a0 = 1 and a1 = a.

  40. USING THE PROPERTIES OF LOGARITHMS Example 4 Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. a. Solution Product property

  41. USING THE PROPERTIES OF LOGARITHMS Example 4 Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. b. Solution Quotient property

  42. USING THE PROPERTIES OF LOGARITHMS Example 4 Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. c. Solution Power property

  43. USING THE PROPERTIES OF LOGARITHMS Example 4 Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. d. Use parentheses to avoid errors. Solution Be careful with signs.

  44. USING THE PROPERTIES OF LOGARITHMS Example 4 Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. e. Solution Power property

  45. USING THE PROPERTIES OF LOGARITHMS Example 4 Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. f. Power property Product and quotient properties

  46. USING THE PROPERTIES OF LOGARITHMS Example 4 Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. f. Solution Power property Distributive property

  47. USING THE PROPERTIES OF LOGARITHMS Example 5 Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. a. Solution Product and quotient properties

  48. USING THE PROPERTIES OF LOGARITHMS Example 5 Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. b. Solution Power property Quotient property

  49. USING THE PROPERTIES OF LOGARITHMS Example 5 Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. c. Solution Power properties

  50. USING THE PROPERTIES OF LOGARITHMS Example 5 Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. c. Solution Product and quotient properties Rules for exponents

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