Section 5: Logarithmic Functions

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 8.5 Logarithmic Functions 821 Version: Fall 2007 8.5 Logarithmic Functions We can now apply the inverse function theory from the previous section to the exponen- tial function. From Section 8.2, we know that the function f ( x ) = b x is either increasing (if b > 1 ) or decreasing (if 0 < b < 1 ), and therefore is one-to-one. Consequently, f has an inverse function f 1 . As an example, let’s consider the exponential function f ( x ) = 2 x . f is increasing, has domain D f = ( −∞ , ) , and range R f = (0 , ) . Its graph is shown in Figure 1 (a). The graph of the inverse function f 1 is a reflection of the graph of f across the line y = x , and is shown in Figure 1 (b). Since domains and ranges are interchanged, the domain of the inverse function is D f 1 = (0 , ) and the range is R f 1 = ( −∞ , ) . x 5 y 5 f y = x x 5 y 5 f 1 y = x (a) (b) Figure 1. The graphs of f ( x ) = 2 x and its inverse f 1 ( x ) are reflec- tions across the line y = x . Unfortunately, when we try to use the procedure given in Section 8.4 to find a formula for f 1 , we run into a problem. Starting with y = 2 x , we then interchange x and y to obtain x = 2 y . But now we have no algebraic method for solving this last equation for y . It follows that the inverse of f ( x ) = 2 x has no formula involving the usual arithmetic operations and functions that we’re familiar with. Thus, the inverse function is a new function. The name of this new function is the logarithm of x to base 2 , and it’s denoted by f 1 ( x ) = log 2 ( x ) . Recall that the defining relationship between a function and its inverse (Property 14 in Section 8.4) simply states that the inputs and outputs of the two functions are interchanged. Thus, the relationship between 2 x and its inverse log 2 ( x ) takes the following form: v = log 2 ( u ) ⇐⇒ u = 2 v More generally, for each exponential function f ( x ) = b x ( b > 0 , b = 1 ), the inverse function f 1 ( x ) is called the logarithm of x to base b , and is denoted by log b ( x ) . The defining relationship is given in the following definition. Copyrighted material. See: 1
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822 Chapter 8 Exponential and Logarithmic Functions Version: Fall 2007 Definition 1. If b > 0 and b = 1 , then the logarithm of u to base b is defined by the relationship v = log b ( u ) ⇐⇒ u = b v . (2) In order to understand the logarithm function better, let’s work through a few simple examples. l⚏ Example 3. Compute log 2 (8) . Label the required value by v , so v = log 2 (8) . Then by ( 2 ), using b = 2 and u = 8 , it follows that 2 v = 8 , and therefore v = 3 (solving by inspection).
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