In order to understand the logarithm function better, let s work through a few simple examples. by (2)

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1 Section 8. Logarithmic Functions Logarithmic Functions We can now appl the inverse function theor from the previous section to the eponential function. From Section 8.2, we know that the function f() = b is either increasing (if b > 1) or decreasing (if 0 < b < 1), and therefore is one-to-one. Consequentl, f has an inverse function f 1. As an eample, let s consider the eponential function f() = 2. 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 =, 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 = (, ). f = = f 1 (a) Figure 1. The graphs of f() = 2 and its inverse f 1 () are reflections across the line =. Unfortunatel, when we tr to use the procedure given in Section 8.4 to find a formula for f 1, we run into a problem. Starting with = 2, we then interchange and to obtain = 2. But now we have no algebraic method for solving this last equation for. It follows that the inverse of f() = 2 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 to base 2, and it s denoted b f 1 () = log 2 (). Recall that the defining relationship between a function and its inverse (Propert 14 in Section 8.4) simpl states that the inputs and outputs of the two functions are interchanged. Thus, the relationship between 2 and its inverse log 2 () takes the following form: v = log 2 (u) u = 2 v More generall, for each eponential function f() = b (b > 0, b 1), the inverse function f 1 () is called the logarithm of to base b, and is denoted b log b (). The defining relationship is given in the following definition. (b) 1 Coprighted material. See:

2 822 Chapter 8 Eponential and Logarithmic Functions Definition 1. If b > 0 and b 1, then the logarithm of u to base b is defined b 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 eamples. Eample 3. Compute log 2 (8). Label the required value b v, so v = log 2 (8). Then b (2), using b = 2 and u = 8, it follows that 2 v = 8, and therefore v = 3 (solving b inspection). In the last eample, note that log 2 (8) = 3 is the eponent v such that 2 v = 8. Thus, in general, one wa to interpret the definition of the logarithm in (2) is that log b (u) is the eponent v such that b v = u. In other words, the value of the logarithm is the eponent! Eample 4. Compute log 10 (10 000). Again, label the required value b v, so v = log 10 (10 000). B (2), it follows that 10 v = , and therefore v = 4. Note that here again we have found the eponent v=4 that is needed for base 10 in order to get 10 v = Eample. Compute log 3 ( 1 9). ) v = log 3 ( 1 9 = 3 v = 1 9 b (2) = v = 2 since 3 2 = 1 9 Eample 6. Solve the equation log () = 1. log () = 1 = 1 = b (2) = =

3 Section 8. Logarithmic Functions 823 Eample 7. Solve the equation log b (64) = 3 for b. log b (64) = 3 = b 3 = 64 b (2) = b = 3 64 = 4 Eample 8. Solve the equation log 1/2 () = 2. = log 1/2 () = 2 ( ) 1 2 = b (2) 2 = = 1 ( 1 2) 2 = = 4 The composition relationships in Propert 1 of Section 8.4, applied to b and log b (), become Propert 9. log b (b ) = (10) and b log b () =. (11) Both equations are important. Note that (11) again shows that the log b () is the eponent v such that b v =. Equation (10) will be used frequentl in this and later sections to help us solve eponential equations. Logarithmic functions are used in man areas of science and engineering. For eample, the are used to define the Richter scale for the magnitudes of earthquakes, the decibel scale for the loudness of sound, and the astronomical scale for stellar brightness. The are also important tools for use in computation (as we will see in Section 8.8). Our main use of logarithms in this tetbook will be to solve eponential equations, and thereb help us stud phsical phenomena that are described b eponential functions (as in Section 8.7). Computing Logarithms In Eamples 3 8 above, we were able to compute the logarithms b converting to eponential equations that could be solved b inspection. But it s eas to see that most of the time this won t work. For eample, how would we compute the value of log 2 (7)?

4 824 Chapter 8 Eponential and Logarithmic Functions Fortunatel, mathematicians have found other methods for computing logarithms to high accurac, and the can now be easil approimated using a calculator or computer. Your calculator has built-in buttons for computing two different logarithms, log 10 () and log e (). log 10 () is called the common logarithm, and log e () is called the natural logarithm. Common Logarithm: The common logarithm log 10 () is computed using the LOG button on our calculator. Notice also that its inverse function 10, can be computed using the same button in conjunction with the 2ND button. The common logarithm is usuall the most convenient one to use for computations involving scientific notation (because we use a base 10 number sstem), and therefore is the logarithm most often used in the phsical sciences. Because of that, it s often just abbreviated b log(), and we ll do that as well in the remainder of the tet. Common Logarithm. log() and log 10 () are equivalent notations. Thus, we have the defining relationship v = log(u) u = 10 v. The composition properties for the common logarithm are and log(10 ) = (12) 10 log() =. Natural Logarithm: The natural logarithm log e () is computed using the LN button on our calculator. Its inverse function, e, is computed using the same button in conjunction with the 2ND button. The natural logarithm turns out to be the most convenient one to use in mathematics, because a lot of formulas, especiall in calculus, are much simpler when the natural logarithm is used. The natural logarithm is abbreviated b ln(). Natural Logarithm. ln() and log e () are equivalent notations. Thus, we have the defining relationship v = ln(u) u = e v. The composition properties for the common logarithm are and ln(e ) = (13) e ln() =.

5 Section 8. Logarithmic Functions 82 Note that when using our calculator to compute log() and ln(), ou will usuall onl obtain approimate values, as these values frequentl are irrational numbers. What about other bases? You can also compute these on our calculator, but we ll first need to develop the Change of Base Formula in the net section. However, at this point, we can at least solve eponential equations involving bases 10 and e, as shown in the net two eamples. Eample 14. Solve the equation 704 = 2(10). The first step is to isolate the eponential on the right side b dividing both sides b 2: 32 = 10 Then simpl appl the log 10 () function to both sides of the equation: log 10 (32) = log 10 (10 ) But (10) implies that log 10 (10 ) =. Therefore, = log 10 (32) = log(32) is the eact solution. The approimate value, using a calculator, is (see Figure 2). Alternativel, instead of taking the logarithm of both sides in the second step, ou can appl (2) to the equation 32 = 10 to get = log 10 (32). Figure 2. Approimation of log(32) = log 10 (32). This last eample shows how logarithms can be used for solving eponential equations. The basic strateg is to first isolate the eponential on one side of the equation, and then take appropriate logarithms of both sides. Here s one more eample for now, and then we ll return to this process repeatedl in the remaining sections, especiall when we work with application problems. Eample 1. Solve the equation 30 = 20e. First isolate the eponential on the right side b dividing both sides b 20: 1. = e This time, since the base of the eponential function is e, appl the natural logarithm function to both sides: log e (1.) = log e (e )

6 826 Chapter 8 Eponential and Logarithmic Functions Simplif the right side, since log e (e ) = b (10): log e (1.) = Therefore, = log e (1.) = ln(1.) is the eact solution. The approimate value, using a calculator, is (see Figure 3). Figure 3. Approimation of ln 1. = log e (1.). In the net section, we ll learn how to solve eponential equations involving other bases. Graphs of Logarithmic Functions At the beginning of this section, we looked at the graphs of f() = 2 and its inverse function f 1 () = log 2 (). More generall, the graph of the eponential function f() = b for b > 1 is shown in Figure 4(a), along with its inverse logarithmic function f 1 () = log b (). According to Section 8.4, the two graphs are reflections across the line =. Similarl, the graph for 0 < b < 1 is shown in Figure 4(b). f = f = f 1 f 1 (a) b > 1 (b) 0 < b < 1 Figure 4. The graphs of f() = b and f 1 () = log b () are reflections across the line =. Because domains and ranges of inverse functions are interchanged, it follows that

7 Section 8. Logarithmic Functions 827 Propert 16. Domain(log b ()) = (0, ) and Range(log b ()) = (, ). In particular, note that the logarithm of a negative number, as well as the logarithm of 0, are not defined. Two particular points on the graph of the logarithm are noteworth. Since b 0 = 1, it follows that log b (1) = 0, and therefore the -intercept of the graph of log b () is (1, 0). Similarl, since b 1 = b, it follows that log b (b) = 1, and therefore (b, 1) is on the graph. Propert 17. log b (1) = 0 and log b (b) = 1 Finall, since the graph of b has a horizontal asmptote = 0, the graph of log b () must have a vertical asmptote = 0. This behavior is a consequence of the fact that inputs and outputs of inverse functions are interchanged, and can be observed in Figure 4. In the final eample below, we ll appl a transformation to the logarithm and see how that affects the graph. Eample 18. Plot the graph of the function f() = log 2 ( + 1). The graph of f() = log 2 ( + 1) will be the same as the graph of g() = log 2 () shifted one unit to the left. The graph of g is shown in Figure 1(b). The -intercept (1, 0) on the graph of g will be shifted one unit to the left to (0, 0) on the graph of f. Likewise, the vertical asmptote = 0 on the graph of g will be shifted one unit to the left to the line = 1 on the graph of f. The final graph of f is shown in Figure. f Figure. The graph of f() = log 2 ( + 1).

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