Logarithmic Functions

Transcription

1 Logarithmic Functions Another common tpe of non-linear function is the logarithmic function. B definition, the logarithmic function is directl related to the eponential function; the two functions are called inverses of one another, much like = ± is the inverse of =. An equation for the inverse of an eponential function can be obtained b switching the places of and as shown below: Eponential Function: = b (Recall that b > 0, b 1) Inverse function: = b (This is called eponential form for a logarithmic function.) Taking this inverse function and solving for, we create a new powerful term. The equation tells us that is the eponent on b that results in. The words logarithm and eponent are interchangeable, so we ma sa that is the logarithm on b that results in. This description can be abbreviated as = log b (This is called logarithmic form for a logarithmic function.) which reads equals log, base b, of or equals log of, base b. For an constant b > 0, b 1, the equation = log b defines a logarithmic function with base b and domain all > 0. Think of eponential form ( = b ) and logarithmic form ( = log b ) as two sides of the ver same coin. Converting a logarithmic function from logarithmic form to eponential form (and vice versa) is helpful for graphing, as we will see in this first eample. Eamples: (1) Graph =. Solution: Rewrite the equation in eponential form, =, and then input values for and find the corresponding value. For eample, when =, = = 8, and when =, = = 1 4 or / 16 or / 8 or / 4 or / or ou ma notice that while the graph of = includes points such as (-1, 0.5), (0, 1) and (1, ), this logarithmic function contains (0.5, -1), (1, 0), and (, 1); this goes back to the idea that if the point represented b the ordered pair (a, b) is on the graph of a function, (b, a) will be on the graph of its inverse. Also, TI calculators have a DrawInv command that enables ou to draw the inverse of an given function. We will learn log rules that will make entering a function like = even more straightforward. 1

2 Net, we plot the points in our table, and then draw a continuous curve that contains the points. = Reviewing the relationship between the two forms, 16 = 4 because 4 = 16; 8 = because = 8; = 1 because 1 = ; 1 = 0 because 0 = 1; and 1 = 1 because 1 = 1. Since the logarithmic function and the corresponding eponential function are inverses, the two graphs are directl related; one is the mirror reflection of the other over the graph of the identit function, =. = = = The chart below shows the critical functions we consider in Sections 5., 5., 5.4, and 5.5. Logarithmic Functions Eponential Functions Logarithmic form Eponential form = log b b = = b These are equivalent forms. Using methods similar to Eample 1, we can graph the logarithmic function = log 10, along with its inverse, = 10. The function = log 10, often abbreviated as = log, is called the common logarithm because of the common use of base 10 in numeration and in metric measurement. [ou ll see the LOG ke on most scientific or graphing calculators, and ou can directl enter = log into our graphing calculator in order to graph this logarithmic function.]

3 And here are the eponential and logarithmic functions, base 10. Points on the graph of = log include (0.01, ), (0.1, 1), (1, 0), (10, 1), and (100, ) We can also graph the logarithmic function = log e together with its inverse, = e. The function = log e, often abbreviated as = ln, is called the natural logarithm because of the use of the base e in several natural applications. [ou ll see the LN ke on most scientific or graphing calculators, and ou can directl enter = ln into our graphing calculator in order to graph this logarithmic function.] Points on the graph of = ln include (0.01, 4.605), (0.1,.0), (1, 0), (, 0.69), (10,.0), and (100, 4.605). Several of these -coordinates are approimations of irrational numbers The three eamples we ve considered are of the form = log b with bases, 10, and e. Notice that whenever b is greater than 1, the graph will increase from left to right. For the sake of completeness, we include the graph of = log 1 1 along with its inverse, = and the identit function, =. = 1 = = log 1 Here is a summar of some of the general features of the graphs of logarithmic functions: The curves are continuous, increasing over their entire domains if b > 1 and decreasing if 0 < b < 1. The domain is { > 0}, and the range includes all real numbers. These facts are important for basic calculations and for solving equations. Each logarithmic function has no -intercept, and their -intercepts are (1, 0) since log b 1 = 0 for an base b. The vertical line = 0 (-ais) is the onl asmptote for the curve. One of the real kes in dealing with logarithms is freel going back and forth between logarithmic form and eponential form.

4 Recall that eponential form ( = b ) and logarithmic form ( = log b ) are equivalent. Eample: () Convert to the other form in each case. (a) 1,04 = 10 Solution: 1,04 = 10 (b) 10 = Solution: log = or log = (c) ln e = 1 Solution: e 1 = e 1 (d) log = 4 Solution: 4 = (e) a r = t Solution: log a t = r There are several powerful log rules that we will find ver helpful for equation solving and applications. All can be proven using basic rules of eponents. Each rule is given below with some of the rationale for the rule and a few eamples of its correct use. In each case, we make the following assumptions: b > 0, b 1, M > 0, and N > 0. 4 Log Rule Eample(s) 1. log b b = 1 because b 1 = b log = 1, log 10 = 1, ln e = 1. log b 1 = 0 because b 0 = 1 log 7 1 = 0, ln 1 = 0, log 1 = 0. log b b n = n because b n = b n log = 1.5, ln e =, log 10 5 = 5 4. b logbn = n because log b n = log b n 4 6 log9 = 6, 10 ln1 = 9, e = 1 5. log b M N = log b M + log b N = 8 4 = 8 + log since b M b N = b M + N log 1 + log 5 = log 1 5 = log 60 M 6. log b b b log log 100 = log N 100 = log 1000 since bm b N = bm N 7. log b M p = p log b M since b M ln 5 = ln 5 7 = ln 5 ln 7 ( ) p = b M p 4 = 4, log 1 10 = 10 log 1 Eamples: Use the log rules to find eact answers, whenever possible. Otherwise, use our calculator to approimate to 4 decimal places. () log 10 8 Solution: 8 (using rule ) (4) log 1 Solution: 0 (using rule ) 4

5 1 (5) 16 Solution: 1 16 = 1 4 = 0 4 = 4 (using rules 6,, and ) (6) 04,51 Solution: (using our trust calculator!) (7) log( ) Solution: No solution (a domain issue for = log ) (8) ln 7.5 Solution:.0149 (again, using our trust calculator!!) (9) 5 log 5 Solution: 5 log 5 = 5 log 5 9 = 9 (using rules 7 and 4) (10) Solution: 1 (using rule 1) There is another etremel important log rule called the change-of-base formula, shown below. Assuming that b > 0, b 1, M > 0, log b M = log a M for an base a > 0, a 1. log a b Note: Common choices of base are 10 and e because the are the bases for LOG and LN kes on a scientific or graphing calculator. Eamples/Solutions: Use our calculator to approimate these to 4 decimal places. log10 1 ln 7 (11) 10 = =. 19 (1) 7 = ln (1) log 40 log 1 40 = 5.19 (14) log 5 5 = log 1 ( ) 5 log 5 or ln 5 ln 5 = This change-of-base formula enables ou to use a graphing calculator to graph = log b for an base b. For eample, we saw the logarithmic function = graphed earlier using the form change technique ( = ) and a table of values. Using the change-of-base formula, we ma write = log ln or =. Entering either of these in the graphing calculator would produce the ln graph we found earlier. Similarl, in order to graph the logarithmic function = log 5 or 5 =, we would use the change-of-base formula to write the equivalent function = log ln or =. log 5 ln 5 All four of these forms describe the same function and curve. In order to prepare for applications involving eponents and logarithms, we need to introduce some equation-solving techniques. 5

6 In each case below, we assume that b > 0, b 1, M > 0, and N > 0. Principles for Eponential and Logarithmic Equation Solving = log b is equivalent to b = Changing from logarithmic form to eponential form, or vice versa, is a ver common and powerful technique. log b M = log b N is equivalent to M = N This is useful both directions dropping the logs or taking the log of both sides of an equation b M = b N is equivalent to M = N This is especiall useful in the direction presented here dropping the identical base, setting the eponents equal. Eamples: Solve for. Be sure to check our solutions. (15) log 5 = Solution: log 5 = is equivalent to = 5. Taking the square root of both sides of the equation, we have = ± 5. Since the base must be positive, the onl solution is = 5. ou can check this with the equivalent form ( 5) = 5 or with the change-of-base formula log 5 = log 5 = 5 log 5. (16) ( 4) = Solution: This is equivalent to = 4. Since = 1 or 1, we have 8 4 = 1 8. Multipling both sides b 1 4, we have = or = 1. ou can check this result with the change-of-base formula or using the equivalent form, as shown: = 4 1 = 1. 8 (17) = 150 Solution: Taking the log (or ln) of both sides, we have = log 150. Using rule 7, we can bring the power in front of : = log 150. Then we can divide both sides of the equation b (or ln ): log150 = This problem can also be handled using the equivalent form and the change-of-base formula. The check involves approimate values: (18) 5 = 1 5 Solution: Since 1 5 = 5 1, we have 5 = 5 1. Drop the base, setting the eponents equal, then solve for : = 1 = 1 To check, substitute: 5 1 = 5 1 =

7 Eamples/Solutions: Solve for. Be sure to check our solutions. (19) ln ( + 1) ln = ln 4 Solution: Combine the left-hand side using rule 6: ln +1 = ln 4. Then drop the logs, and solve the proportion using cross products and linear equation-solving techniques. +1 = = 4 1 = = 1 1 The check is straightforward in the calculator: 1 1 ln + 1 ln 1.86 and ln (0) log = 5 Solution: B rule 7, we ma write log = 5, then we can drop the logs, and use the square root propert. = 5 = ± 5 = ±5 Because of the domain of logarithmic functions, = 5 is the onl solution, and the check is log and Eercises: 1. Complete the chart, converting the given equation from one form to the other. Logarithmic form Eponential form (a) log 81 = 4 (b) log = (c) 7 5 = 16,807 (d).9957 e 0 (e) log 5 1 = 0 (f) 1 1 = 1 (g) ln Evaluate. Give eact answers, whenever possible. (a) 64 (b) log 100,000,000 (c) ln ( e 4 ) (d) log (-10) (e) log Evaluate. Give eact answers, whenever possible. (a) log 1 (b) ln 0 (c) log 4 ( 1 ) (d) log 1 4 (e) log

8 4. (a) Complete the tables below for the given eponential and logarithmic functions. = = log (b) Graph these two functions along with = on the coordinate grid. Include an asmptote(s) and intercept(s). 10 (c) What do ou notice about the tables in part (a)? (d) What do ou notice about the graphs? For the logarithmic function given b = ( + ) (a) Complete the table below, and graph the function over an appropriate domain. Include an asmptote(s) and intercept(s) (b) How does this graph compare with = in shape and location on the coordinate grid? Refer to the graph of = that we developed earlier in this section. 8

9 6. Evaluate. Use our calculator and/or the log rules to approimate these to 4 decimal places. (a) log 15 (b) ln 44 (c) log 7 8 (d) log (e) log π e (f) log 7 log 7 (g) 8 log (h) log Solve for. Give eact answers, if possible. Otherwise, use our calculator to give approimate solutions correct to 4 decimal places. (a) log = 10 (b) log 10 = (c) 1 = 9 (d) ( + 1) + ( 1) = (e) 5 = 0 8. Solve for. Give eact answers, if possible. Otherwise, use our calculator to give approimate solutions correct to 4 decimal places. (a) log( + 1) = log 8 (b) 4 = 7 (c) log 4 ( 5 ) = log 4 ( + 7) (d) ln ( 1) ln ( ) = ln (e) 10 = Find the value of log 4 log 4 5 log 5 6 log 6 7 log Find the value of 4 log 4 6 log 6 8 log Write each epression as a single logarithm. (a) log u + log v (b) log u 1 log v 1. Write each epression as a sum and/or difference of logarithms. Epress powers as factors. (a) 4 ( ) (b) log 5 (c) log ( ) (d) ln(e) 9