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1 6.4 Logarithmic Functions In this section, we will study the following topics: Evaluating logarithmic functions with base a Graphing logarithmic functions with base a Evaluating and graphing the natural logarithmic function Solving logarithmic and exponential equations

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2 Logarithmic Functions Now that you have studied the exponential function, it is time to take a look at its INVERSE: THE LOGARITHMIC FUNCTION. In the exponential function, the independent variable (x) was the exponent. So we substituted values into the exponent and evaluated it for a given base. Exponential Function: f(x) = 2 x, f(3) = 2 3 = 8.

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3 Logarithmic Functions For the inverse function (LOGARITHMIC FUNCTION), the base is given and the answer is given, so to evaluate a logarithmic function is to find the exponent. That is why I think of the logarithmic function as the “Guess That Exponent” function. Warm Up: Give the value of ? in each of the following equations.

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4 Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Subliminal Message

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5 Logarithmic Functions (continued) Evaluate log 2 8 To evaluate log 2 8 means to find the exponent such that 2 raised to that power gives you 8.

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6 Logarithmic Functions (continued) The following definition demonstrates this connection between the exponential and the logarithmic function. Definition of a Logarithmic Function For x > 0, a > 0, and a ≠ 1, y = log a x if and only if x = a y We read log a x as “log base a of x”.

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7 Converting Between Exponential and Logarithmic Forms I. Write the logarithmic equation in exponential form. a) b) II. Write the exponential equation in logarithmic form. a) b) y = log a x if and only if x = a y

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8 Evaluating Logarithms w/o a Calculator To evaluate logarithmic expressions by hand, we can use the related exponential expression. Example: Evaluate the following logarithms:

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9 Evaluating Logarithms w/o a Calculator (cont.)

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10 Evaluating Logarithms w/o a Calculator Okay, try these. e) f) g) h)

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11 The Common Logarithm The common logarithm has a base of 10. If the base of a logarithm is not indicated, then it is assumed that the base is 10.

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12 Graphs of Logarithmic Functions Since the logarithmic function is the _______________ of the exponential function (with the same base), we can use what we know about inverse functions to graph it. Example: Graph f(x) = 2 x and g(x) = log 2 x in the same coordinate plane. To do this, we will make a table of values for f(x)=2 x and then switch the x and y coordinates to make a table of values for g(x).

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13 Graphs of Logarithmic Functions (continued) f(x) = 2 x g(x) = log 2 x xf(x) -4 -2 0 2 4 xg(x) f(x) = 2 x g(x)= log 2 x y =x Inverse functions

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14 Comparing the Graphs of Exponential and Log Functions Notice that the domain and range of the inverse functions are switched. The exponential function has domain (- , ) range (0, ) HORIZONTAL asymptote y = 0 The logarithmic function has domain (0, ) Range (- , ) VERTICAL asymptote x = 0

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15 Transformations of Graphs of Logarithmic Functions The same transformations we studied earlier also apply to logarithmic functions. Look at the following shifts and reflections of the graph of f(x) = log 2 x. The new vertical asymptote is x = -2

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16 Transformations of Graphs of Logarithmic Functions

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17 The Natural Logarithmic Function In section 6.3, we saw the natural exponential function with base e. Its inverse is the natural logarithmic function with base e. Instead of writing the natural log as log e x, we use the notation, which is read as “the natural log of x” and is understood to have base e.

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18 Natural Log Key To evaluate the natural log using the TI-83/84, use the button. Notice, the 2nd function of this key is ________.

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19 Graph of the Natural Exponential and Natural Logarithmic Function f(x) = e x and g(x) = ln x are inverse functions and, as such, their graphs are reflections of one another in the line y = x.

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20 Evaluate without using a calculator. a) b) c)d) e)f) Evaluating the Natural Log

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21 Solving Logarithmic Equations Strategy for solving logarithmic equations: Change the equation from a log equation into an exponential equation, using one of the following forms: log a x = y x = a y logx = y x = 10 y lnx = y x = e y Keep in mind that the domain of the log function is x>0. Reject any extraneous solutions!!

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22 Examples of solving log equations

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23 More examples of solving log equations

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24 End of Section 6.4

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