multiplication and division, and through these to addition and subtraction. For this labor-saving device we are indebted to John Napier (1550-1617), who arrived at it by considering a function which increased in arithmetical progression as the variable decreased in geometrical progression. Following Napier, this tool was put in a more serviceable form by Henry Briggs (1561-1631). We shall use a table of logarithms for simplifying many computations, especially in connection with the solution of triangles. The introduction of the exponential and logarithmic functions completes the list of functions to be studied in this course (see Classification, page 38). 76. Graph of the Exponential Function b, b>1. In the following table of values of the exponential function 2o, the value of the function for x O is obtained by means of the definition bo 1, and the value for any negative value of x is = = x - 2, 2x -7 6 3 2 1 - 1, 0, 1, 2, -2 -1 0 2 1 FIG. 126. 2, 1, 2, 4, 1 -n = bn 1, and by the second 2-2 3 The table of values is readily computed and the graph plotted as usual. The process is so simple that it is hardly worth while to discuss the graph in advance. The following properties of the graph are apparent. 1. The intercept on the y-axis is 1. 2. The graph lies entirely above the x-axis. 3. The x-axis is an asymptote. 4. Any line parallel to the y-axis cuts the graph once and once only. 5. The graph rises to the right more and more rapidly. The form of the graph of the exponential function b2, b>1, is very much like that of 2. As indicated in Fig. 127, the graph always cuts the y-axis at y = 1, lies entirely above the x-axis, which is an asymptote, and runs up to the right. If b>2, the curve rises more rapidly to the right of the y-axis than in the case of 2, and the larger the value of b the more rapid is the rise. -10 9 10 32 The values of b of most importance in mathematics are 10 and a number denoted by "e" whose value to four figures is e = 2.718. It was proved in 1844 that there are numbers which are not the roots of any algebraic equation no matter how high the degree, and the first number definitely proved to be of this sort was e, in 1873. In 1882 this same fact was proved about π, and by means of this it was proved that it is impossible to "square the circle" with ruler and compass. A table of values of e* is given in Huntington's Tables, page 30. -2 2 3 4 x FIG. 127. 77. Properties of the Exponential Function b', b>1. The properties of the function which correspond to the above properties of the graph are as follows: 1. For all exponential functions bo = 1. 2. The exponential function be is positive for all real values. of x. 3. As. x decreases indefinitely through negative values b± approaches 0 as a limit. 4. For each value of x there is one and only one value of ba. 5. The function be increases as x increases and the rate of change of b2 also increases. Of two exponential functions the one with the larger base has the greater rate of change for x>0. Other important properties of the function which are not so apparent from the graph are the following: 6. If the values of x be chosen in arithmetical progression, the corresponding values of the function are in geometrical Irogression. which form a geometrical progression, for any one of these values may be obtained from the preceeding by multiplying by bd. Properties 7, 8, 9, proved in elementary algebra for integral values m and n of x, are true for all values of x. The definition 10 holds also for all values of x. If f(x) = b2, the last four properties may be written, in the reverse order: In these forms, the analogy of these relations respectively to (1), (2), (3), and (4) on page 153 is apparent. Thus seemingly unrelated rules of elementary algebra are in reality analogous properties of the functions be and x2. -7 78. Computation by means of an Exponential Function. The computations in the following examples are considerably simplified by means of the adjoining table and properties 7, 8, 9, 10 of the exponential function. 2x 0.0078125 EXAMPLE 1. Find the product (0.0078125) (1048576). 1048576 220 Whence, by property 7, the product Therefore = = 2-7+20 213 = (0.0078125) (1048576) 8192. EXAMPLE 2. Find the quotient 32768/524288. = = 86 = 262144. 1048576. 11 2048 These examples are sufficient to show that an extensive table would shorten very appreciably the labor of finding a product, quotient, power, or root. In practice it is found more convenient to use a table of values of the inverse function. EXERCISES 1048576 = V/220. = 220/5 = 1048576 16. = 1. Plot the graphs of 2*, e*, and 3* on the same set of axes. (See page 30 of the Tables for values of e*.) Calculate for each function the value of Ay/Ax for the intervals 2 to 1 to 0, 0 to +1, +1 to + 2, and determine which has the greatest value of ▲y/Ax and which the least for each interval. 2. Plot the graph of 2-* from a table of values. In what two ways is the graph related to that of 2*? 3. How does 3- change as x increases in arithmetic progression? 4. Plot the graphs of the following functions on the same axes 2o, (§)*, (1)*, (3), ()*. What two relations exist between the graphs of the second and the fourth function? How does the function b vary if b> 1? if b = 1? if b < 1? 5. Plot the graphs of e*, e ̄*, (e* + e ̄*) and (e* - e-*) on the same axes for the range x - 2 to x = +2 at intervals of 0.5. (See Tables, p. 30.) = The functions (e + e-*) and (e* - e-* ), have somewhat the same relation to the equilateral hyperbola that the trigonometric functions have to the circle. They are called respectively the hyperbolic cosine and hyperbolic sine and are symbolized by cosh x and sinh x. 6. If x', y' and x", y" are two sets of values satisfying the equation y = b2, show that the value of the function corresponding to the arithmetic mean of x' and x" is the geometric mean of y' and y". 7. Find the value of 2* for x = 1.50, using Exercise 6 and the values for x = 1 and x 2. Then find the value of 2 for x Arrange the results in tabular form. = = 1.25; for x = 1.75. 8. Referring to the table in Exercise 7, between what two values of x does x = ✓2 lie? The value of 21.41 is an approximate value of 22. Find it by applying Exercise 6 several times, using the table in Exercise 7, and choosing the successive values of x' and x" so that their arithmetic mean approaches 1.41. 9. By means of the table in Section 78 find the value of each of the following: (a) (524288) (0.015125). (d) √16384. (c) 164. (b) 4096/0.0078125. 10. A glass marble falls from a height of 4 feet and rebounds one-half the distance fallen, falls again and rebounds one-half the preceding distance fallen, and so on. Express the distance fallen each time as a function of the number of times it has fallen and draw the graph. How far does it fall the eighth time? 11. If the planets are numbered in the order of their distance from the sun the distance from the sun to the nth planet is approximately 4 + 3(2)"−2, the distance of the earth being represented by 10. Compare the results of substituting in this formula with the following table. If there is a planet external to Neptune, at what approximate distance may we expect it to be found? |