CHAPTER 8319CHAPTER TABLE OFCONTENTS8-1Inverse of an Exponential Function8-2Logarithmic Form of an Exponential Equation8-3Logarithmic Relationships8-4Common Logarithms8-5Natural Logarithms8-6Exponential Equations8-7Logarithmic EquationsChapter SummaryVocabularyReview ExercisesCumulative ReviewLOGARITHMIC FUNCTIONSThe heavenly bodies have always fascinated and challenged humankind. Our earliest records contain conclusions, some false and some true, that were believed about the relationships among the sun, the moon, Earth, and the other planets.As more accurate instruments for studying the heavens became available and more accurate measurements were possible, the mathematical computations absorbed a great amount of the astronomers time. A basic principle of mathematical computation is that it is easier to add than to multiply. John Napier (15501617) developed a system of logarithms that facilitated computation by using the principles of expo- nents to find a product by using addition.Henry Briggs (15601630) developed Napiers concept using base 10. Seldom has a new mathematical concept been more quickly accepted.14411C08.pgs 8/12/08 1:51 PM Page 319In Chapter 7, we showed that any positive real number can be the expo- nent of a power by drawing the graph of the exponential function y 5 bxfor 0 , b , 1 or b . 1. Since y 5 bxis a one-to-one function, its reflection in the line y 5 x is also a function. The function x 5 byis the inverse function of y 5 bx.b . 10 , b , 1The equation of a function is usually solved for y in terms of x.To solve the equation x = byfor y, we need to introduce some new terminology. First we will describe y in words:x 5 by:“y is the exponent to the base b such that the power is x.”A logarithm is an exponent.Therefore, we can write:x 5 by:“y is the logarithm to the base b of the power x.”The word logarithm is abbreviated as log. Look at the essential parts of this sentence:y 5 logbx:“y is the logarithm to the base b of x.”The base b is written as a subscript to the word “log.”?x 5 bycan be written as y 5 logbx.For example, let b 5 2.Write pairs of values for x 5 2y and y 5 log2x.xyy = xy = bxx = byOxyOx = byy = xy = bx8-1 INVERSE OF AN EXPONENTIAL FUNCTION320Logarithmic Functions14411C08.pgs 8/12/08 1:51 PM Page 320We say that y 5 logbx, with b a positive number not equal to 1, is a logarithmic function.EXAMPLE 1Write the equation x 5 10yfor y in terms of x.Solutionx 5 10y y is the exponent or logarithm to the base 10 of x.y 5 log10xGraphs of Logarithmic FunctionsFrom our study of exponential functions in Chapter 7, we know that when b . 1 and when 0 , b , 1, y 5 bxis defined for all real values of x.Therefore, the domain of y 5 bxis the set of real numbers.When b . 1,as the negative val- ues of x get larger and larger in absolute value, the value of bxgets smaller but is always positive. When 0 , b , 1, as the positive values of x get larger and larger,the value of bxgets smaller but is always positive.Therefore,the range of y 5 bxis the set of positive real numbers. When we interchange x and y to form the inverse function x 5 byor y 5 logbx:?The domain of y 5 logbx is the set of positive real numbers.?The range y 5 logbx is the set of real numbers.?The y-axis or the line x 5 0 is a vertical asymptote of y 5 logbx.Inverse of an Exponential Function321x 5 2yIn Wordsy 5 log2x(x, y)5 211 is the logarithm to the base 2 of .1 5 200 is the logarithm to the base 2 of 1.0 5 log21(1, 0)is the logarithm to the base 2 of .2 5 211 is the logarithm to the base 2 of 2.1 5 log22(2, 1)4 5 222 is the logarithm to the base 2 of 4.2 5 log24(4, 2)8 5 233 is the logarithm to the base 2 of 8.3 5 log28(8, 3)A!2, 12B1 25 log2 !2!21 2!2 5 21 2A1 2, 21B21 5 log2 121 21 214411C08.pgs 8/12/08 1:51 PM Page 321EXAMPLE 2a. Sketch the graph of f(x) 5 2x.b. Write the equation of f21(x) and sketch its graph.Solutiona. Make a table of values for f(x) 5 2x, plot the points, and draw the curve.b. Let f(x) 5 2x y 5 2x.To write f21(x), interchange x and y.x 5 2yis written as y 5 log2x.Therefore, f21(x) 5 log2x.To draw the graph, interchange x and y in each ordered pair or reflect the graph of f(x) over the line y 5 x. Ordered pairs of f21(x) include , (1, 0), (2, 1), (4, 2), and (8, 3).The function y 5 logbx represents logarithmic growth. Quantities represented by a logarithmic function grow very slowly. For example, suppose that the time it takes for a computer to run a program could by modeled by the logarithmic func- tion y 5 log10x where x is the number of instructions of the computer program and y is the running time in milliseconds. The graph on the right shows the running time in the interval 1 , x , 1,000,000. Note that the graph increases from 0 to 5 for the first 100,000 instructions but only increases from 5 to 6 as the number of instructions increases from 100,000 to 1,000,000. (Each interval on the x-axis represents 100,000 instructions.)A1 2, 21BA1 4, 22B322Logarithmic Functionsf(x) ? 2xO