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4.3 - 1 10 TH EDITION LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA

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

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4.3 - 3 Logarithms The previous section dealt with exponential functions of the form y = a x 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.

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4.3 - 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 = a x and interchanging x and y yields

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4.3 - 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 log a 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.”

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4.3 - 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 log a x represents the exponent to which the base “a” must be raised in order to obtain x.

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4.3 - 7 Logarithms Logarithmic form: y = log a x Exponent Base Exponential form: a y = x Exponent Base

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4.3 - 8 Logarithms

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4.3 - 9 Example 1 SOLVING LOGARITHMIC EQUATIONS Solve Solution a. Write in exponential form. Take cube roots

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4.3 - 10 Example 1 SOLVING LOGARITHMIC EQUATIONS Check: ? Let x = ⅔. ? Write in exponential form Original equation True The solution set is

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4.3 - 11 Example 1 SOLVING LOGARITHMIC EQUATIONS Solve Solution b. Write in exponential form. The solution set is {32}.

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4.3 - 12 Example 1 SOLVING LOGARITHMIC EQUATIONS Solve Solution c. Write in exponential form. The solution set is Write with the same base. Power rule for exponents. Set exponents equal. Divide by 2.

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4.3 - 13 Logarithmic Function If a > 0, a ≠ 1, and x > 0, then defines the logarithmic function with base a.

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4.3 - 14 Logarithmic Function Exponential and logarithmic functions are inverses of each other. The graph of y = 2 x is shown in red. The graph of its inverse is found by reflecting the graph across the line y = x.

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4.3 - 15 Logarithmic Function The graph of the inverse function, defined by y = log 2 x, shown in blue, has the y-axis as a vertical asymptote.

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4.3 - 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.

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4.3 - 17 Domain: (0, ) Range: (– , ) LOGARITHMIC FUNCTION x (x)(x) ¼ – 2 – 2 ½– 1– 1 10 21 42 83 (x) = log a x, a > 1, is increasing and continuous on its entire domain, (0, ). For (x) = log 2 x:

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4.3 - 18 Domain: (0, ) Range: (– , ) LOGARITHMIC FUNCTION x (x)(x) ¼ – 2 – 2 ½– 1– 1 10 21 42 83 The y-axis is a vertical asymptote as x 0 from the right. For (x) = log 2 x:

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4.3 - 19 Domain: (0, ) Range: (– , ) LOGARITHMIC FUNCTION x (x)(x) ¼ – 2 – 2 ½– 1– 1 10 21 42 83 The graph passes through the points For (x) = log 2 x:

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4.3 - 20 Domain: (0, ) Range: (– , ) LOGARITHMIC FUNCTION x (x)(x) ¼ 2 ½1 10 2 – 1– 1 4 – 2– 2 8 – 3– 3 (x) = log a x, 0 < a < 1, is decreasing and continuous on its entire domain, (0, ). For (x) = log 1/2 x:

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4.3 - 21 Domain: (0, ) Range: (– , ) LOGARITHMIC FUNCTION x (x)(x) ¼ 2 ½1 10 2 – 1– 1 4 – 2– 2 8 – 3– 3 For (x) = log 1/2 x: The y-axis is a vertical asymptote as x 0 from the right.

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4.3 - 22 Domain: (0, ) Range: (– , ) LOGARITHMIC FUNCTION x (x)(x) ¼ 2 ½1 10 2 – 1– 1 4 – 2– 2 8 – 3– 3 For (x) = log 1/2 x: The graph passes through the points

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4.3 - 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 (– , ).

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4.3 - 24 Example 2 GRAPHING LOGARITHMIC FUNCTIONS Graph the function. Solution a. First graph y = (½) x which defines the inverse function of , by plotting points. The graph of (x) = log 1/2 x 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.

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4.3 - 25 Example 2 GRAPHING LOGARITHMIC FUNCTIONS Graph the function. Solution a.

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4.3 - 26 Example 2 GRAPHING LOGARITHMIC FUNCTIONS Graph the function. Solution b. Another way to graph a logarithmic function is to write (x) = y = log 3 x in exponential form as x = 3 y.

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4.3 - 27 Example 2 GRAPHING LOGARITHMIC FUNCTIONS Graph the function. Solution a.

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4.3 - 28 Caution If 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.

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4.3 - 29 Example 3 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution a. The graph of (x) = log 2 (x – 1) is the graph of (x) = log 2 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 = log 2 (x – 1).

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4.3 - 30 Example 3 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution a. Write in exponential form. Add 1.

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4.3 - 31 Example 3 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution a. We choose values for y and then calculate each of the corresponding x- values. The range is (– , ).

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4.3 - 32 Example 3 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution b. The function defined by (x) = (log 3 x) – 1 has the same graph as g (x) = log 3 x translated 1 unit down. We find ordered pairs to plot by writing y = (log 3 x) – 1 in exponential form.

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4.3 - 33 Example 3 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution b. Add 1. Write in exponential form.

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4.3 - 34 Example 3 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution b. Again, choose y- values and calculate the corresponding x- values. The domain is (0, ) and the range is (– , ).

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4.3 - 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.

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4.3 - 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

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4.3 - 37 Properties of Logarithms For x > 0, y > 0, a > 0, a ≠ 1, and any real number r: Property Quotient Property Description The logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numbers.

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4.3 - 38 Properties of Logarithms For x > 0, y > 0, a > 0, a ≠ 1, and any real number r: Property Power Property Description The logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number.

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4.3 - 39 Properties of Logarithms Two additional properties of logarithms follow directly from the definition of log a x since a 0 = 1 and a 1 = a.

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4.3 - 40 Example 4 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution a. Product property

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4.3 - 41 Example 4 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution b. Quotient property

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4.3 - 42 Example 4 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution c. Power property

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4.3 - 43 Example 4 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution d. Use parentheses to avoid errors. Be careful with signs.

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4.3 - 44 Example 4 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution e. Power property

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4.3 - 45 Example 4 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. f. Power property Product and quotient properties

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4.3 - 46 Example 4 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution f. Power property Distributive property

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4.3 - 47 Example 5 USING THE PROPERTIES OF LOGARITHMS Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. Solution a. Product and quotient properties

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4.3 - 48 Example 5 USING THE PROPERTIES OF LOGARITHMS Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. Solution b. Power property Quotient property

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4.3 - 49 Example 5 USING THE PROPERTIES OF LOGARITHMS Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. Solution c. Power properties

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4.3 - 50 Example 5 USING THE PROPERTIES OF LOGARITHMS Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. Solution c. Product and quotient properties Rules for exponents

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4.3 - 51 Example 5 USING THE PROPERTIES OF LOGARITHMS Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. Solution c. Rules for exponents Definition of a 1/n

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4.3 - 52 Caution There is no property of logarithms to rewrite a logarithm of a sum or difference. That is why, in Example 5(a), log 3 (x + 2) was not written as log 3 x + log 3 2. Remember, log 3 x + log 3 2 = log 3 (x 2). The distributive property does not apply in a situation like this because log 3 (x + y) is one term; “log” is a function name, not a factor.

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4.3 - 53 Example 6 USING THE PROPERTIES OF LOGARITHMS WITH NUMERICAL VALUES Assume that log 10 2 =.3010. Find each logarithm. Solution a. b.

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4.3 - 54 Theorem on Inverses For a > 0, a ≠ 1:

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4.3 - 55 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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