12. Show for the exponential function e* that the average rate of change Find the value of the second factor if Ax= 0.1, 0.01. Assuming that the limit of the second part is 1 as Ax approaches 0, find m for the exponential e* and determine the angle that the graph makes with the y-axis. = = 13. Plot the graph of y kemx, first letting k cessively equal to – 2, − 1, 0, + 1/1, + 1, + 2, and choosing k successively equal to − 1, +1, +2. and k affect the graph? 14. The pull, P, needed to check a weight, W, being lowered by means of a rope wrapped around an iron druin, is given by the equation P = We-me, where the coefficient of friction is m = 0.3, and 0, the angle of contact of the rope with the drum, is measured in radians. If W 500 pounds plot the graph of P as a function of 0. What is the value of P if the rope is wrapped 4 times around the drum? = 79. The Logarithmic Function, the Inverse of the Exponential Function. To find the inverse of 2*, let y = 2 and interchange x and y, obtaining x = 2. The solution of this equation for y in terms of x is a new function called the logarithm of x to the base 2, and it is denoted by log2 x. Thus = X 24 and y = log2 x are forms of the same equation, in the one case solved for x and in the other for y. In general, we have the DEFINITION. If bmn, then m is said to be the logarithm of n to the base b. Hence the logarithm of a number to the base b is the exponent of the power to which b must be raised to equal the given number. It is frequently convenient to change from the exponential to the logarithmic form of this relation, and vice-versa. For example, from 35 243, we have logs 243 = 5. = = = Again, to find logs 625, we let logs 625 m, whence 5m 625. Since 54 625, we have m = 4 and hence logs 625 = 4. = 80. Graph of the Logarithmic Function. The graph of y = log x may be readily obtained from that of 2 by the rule in Section 39, page 113, and from it certain properties of the graph are obvious. 2. The graph lies entirely to the right of the y-axis. 4 FIG. 128. 5 6 x 5. The graph rises to the right but at a decreasing rate. 6. A line parallel to the x-axis with an intercept on the y-axis at y = 1 cuts the graph at a point whose abscissa is x = 2. As indicated in Fig. 129, which shows the graph of log, x for b = 2, 3, 10, the graph of any logarithmic function, y = log, x, b>1, cuts the x-axis at x = 1, lies entirely to the right of the y-axis, Corresponding to the properties of the graph are the following properties of the function: = 1. 1. For any base, log 10. For bo 2. Log, x is a real number for positive values of x only. 3. As x approaches 0, log, x approaches ∞, that is, 4. The logarithm of a positive number greater than unity is positive. The logarithm of a positive number less than unity is negative. 5. The function log, x increases and the rate of change of log x decreases as x increases, and of two logarithmic func tions, the one with the larger base has a smaller rate of change for x>1. 6. The logarithm of the base itself is unity. For since b1 = b, we have log b = 1. Other properties of the function are easily deduced from the corresponding properties of the exponential function as follows: 7. Theorem. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. 8. Theorem. The logarithm of the quotient of two numbers is equal to the logarithm of the dividend minus the logarithm of the divisor. Let p = bTM and q = b", whence log, p = m and log, q = n. Then p/q = bm /bn = bm-n 9. Theorem. The logarithm of the nth power of a number equals n times the logarithm of the number. 10. Theorem. The logarithm of the nth root of a number equals the logarithm of the number divided by n. If f(x) = logɩx, properties 7, 8, 9, 10, may be written These properties are respectively analogous to those of x" given by (3), (4), (5), (6), page 153. If the first two be written in the reverse order, we have which are analogous to (7) and (8), page 153. By means of the theorems in 7 and 9 the solutions of Examples 1 and 3, page 219, may be written as follows: 1. Express the following exponential equations in logarithmic form. 2. What are the logarithms of 1, 2, 16, 1024, 0.125 with respect to the base 2? Express the answers in exponential and logarithmic form. 3. Find log2 2048, logs 81, loge 2.718, log 9, log√ 16. Express the answers in exponential form. 4. Find log, b, blog, log (log, 16), log. (1/e), blog-log. 5. (a) Change the equation loge = (1 − x) — 2logʊ x − logь (1 + x). kt + log. Oo to the form 0 = 0, e-kt. (b) Show that - log (1/x2) = log, x. (c) Find f(x) if log, f(x) = log 6. Plot the graphs of log2 x, loge x, logs x, page 31 of the Tables for values of log. x.) for each function for the interval 1 to 2. on the same set of axes. (See Calculate the value of Ay/Ax 7. Plot the graph of log1/ex. In what two ways can this graph be obtained from that of log. x? 8. Illustrate each of the theorems in 7, 8, 9, 10, Section 81, on the graph of log2 x. 9. Write the solutions of Examples 2 and 4, Section 78, in logarithmic forms. 10. Show by Theorems 7 and 8 that log = logь p + logь q − logɩ r. r 11. Using the table in Section 78 find (0.03125) (262144)/(32768). 12. Prove the theorem, if the numbers x, x', x', etc., are in geometric progression, their logarithms are in arithmetic progression. 13. Plot the graphs of log: x, log; x2, log:√x on the same axes. = = How can the graphs of the last two be obtained from that of the first function? 14. Plot the graph of y log2 (x + k) for the values k 0. 1, 1 on the same axes, and discuss the change effected in the graph by a change in k. 15. The equation for the economic law of diminishing utility is k log (x/c), where x denotes income; y happiness; c the income sufficient. for necessities; and k is a constant depending on individuality. Plot a graph of the law, determine the graphical significance of c and k, and discuss the law for xc. y 82. Common Logarithms. Logarithms to the base 10 are called common logarithms. The first table of logarithms to the base 10 was published in 1617 by Henry Briggs. It contained logarithms of numbers from 1 to 1000. In writing logarithms to the base 10, the base is usually omitted. Thus y = log10 x is written y = log x, which means that x 10. That is = The common logarithm of a number is the power to which 10 must be raised to obtain the number. A four-place table of common logarithms of numbers from 1 to 10 is given on pages 16 and 17 in Huntington's Tables. The graph of this table of logarithms is shown in the figure. The method of interpolation is the same as in the tables already used. W 1+ P 1 The number 2.342, and its logarithm, the number 0.3696, are the coördinates of a point P on the graph. |