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4.3

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

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4.3

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  1. 4.3 Logarithmic Functions Logarithms Logarithmic Equations Logarithmic Functions Properties of Logarithms

  2. Logarithms The previous section dealt with exponential functions of the form y = ax where a ≠ 1 and a>0. The horizontal line test shows that exponential functions are one-to-one, and thus have inverse functions. Starting with y = ax and interchanging x and y yields x = ay .Here we need to solve for y

  3. Logarithms Here y is the exponent. We call this exponent a logarithm, symbolized by “log.” We wrte y=loga x . 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.”

  4. 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.

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

  6. Logarithms

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

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

  9. 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.

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

  11. 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.

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

  13. Logarithmic Function

  14. LOGARITHMIC FUNCTION For (x) = log2x: • (x) = logax, a > 1, is increasing and continuous on its entire domain, (0, ) . • The y-axis is a vertical asymptote as x  0 from the right.

  15. LOGARITHMIC FUNCTION For (x) = log1/2x: • (x) = logax, 0 < a < 1, is decreasing and continuous on its entire domain, (0, ) . • The y-axis is a vertical asymptote as x  0 from the right.

  16. GRAPHING LOGARITHMIC FUNCTIONS Example 2 First graph y = (½)x . The graph of (x) = log1/2x is the reflection of the graph y = (½)x across the line y = x. Graph the function. a. Solution

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

  18. 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).

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

  20. 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.

  21. 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 (–, ).

  22. 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

  23. 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

  24. 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

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

  26. 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

  27. 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

  28. 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

  29. 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.

  30. 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

  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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 Rules for exponents Definition of a1/n

  38. CautionThere is no property of logarithms to rewrite a logarithm of a sum or difference. That is why, in Example 5(a), log3(x + 2) was not written as log3x + log3 2. Remember, log3x + log3 2 = log3(x • 2). The distributive property does not apply in a situation like this because log3 (x + y)is one term; “log” is a function name, not a factor.

  39. USING THE PROPERTIES OF LOGARITHMS WITH NUMERICAL VALUES Example 6 Assume that log10 2 = .3010. Find each logarithm. a. Solution b.

  40. Theorem on Inverses For a > 0, a ≠ 1:

  41. Theorem on Inverses By the results of this theorem, The second statement in the theorem will be useful in Sections 4.5 and 4.6 when we solve other logarithmic and exponential equations.

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