3. At Berlin and Cape of Good Hope the zenith distance z1 and z2 of the moon were measured simultaneously and found to be zi = 53° 10' 1", Find the distances BM and OM. 4. When the moon is in the position where we see half the illuminated portion (i.e. half-moon) the angle at the moon is 90°. At such an instant the angle at E was measured and found to be 89.86°. Calculate the distance to the sun using the distance to the moon as a base line. 5. The annual parallax of a star is the angle at the star subtended by the semi-diameter of the earth's orbit. Observations of Jupiter taken from diametrically opposite points of the earth's orbit give the annual parallax of Jupiter as 11° 4′ 58′′. Find the distance of the sun from Jupiter. per second, find the length of time necessary for light to traverse the distance. 7. The annual parallax of the North Star is 0.073". Find the distance to the North Star in light-years. 1 Inspection of the Condensed Table on the inside of the back cover of the Tables shows that values of 0, in radians, and of tan agree to three decimal places if 0 ≤ 6°; to four places if 0 ≤ 3°. 8. Find the time that it takes light to reach the earth from the sun; to reach Neptune, the most remote planet, distance 2,788,000,000 miles. 85. Exponential Equations. An equation in which the unknown enters as an exponent is called an exponential equation. Exponential equations of the type af(x) = bF(x), where f(x) and F(x) are linear or quadratic algebraic functions may be solved by equating the logarithms of the two sides of the equation and solving the resulting algebraic equation, as in the following examples. = EXAMPLE 1. Find the function of the type y x" whose graph passes through the point (3, 5). Substituting the coördinates of the point in the equation, we have 3n = 5. Equating the logarithms of both sides of this equation, we have The value of n is here calculated from the quotient by logarithms rather than by long division. the result would be log Notice that if 0.4771 were subtracted from 0.6990, (5/3) and not log 5/log 3. EXAMPLE 2. Find the logarithm to the base e of any positive number x assuming a table of common logarithms to be given. Let Equating the common logarithms of both sides of the second equation we have That is, the logarithm to the base e of any number can be found by multiplying the common logarithm of the number by 1/0.4343. is called the modulus of the base e with respect to the base 10. The figure shows the graphs of log10 x and log. x. OA = e, then AB = log10 e. loge x = PR. Hence equation (1) may be written in the form PR บ 3 2 R logex F 2.3026 D 2 e3 0.4343 5 8 9 10 -2· FIG. 139. AB 0.4343 and the graph of log, x can be obtained from that of log10 x by the theorem on page 89. EXAMPLE 3. The variation of atmospheric pressure p (in millimeters of mercury in a barometer) with respect to the altitude h above sea level (in meters) is given by the equation p = Poe-mh, where po and m are con stants and e is the base of the natural system of logarithms. If p = 719 when h 450, and p 594 when h 2000, find p if h = 8000. = Before starting the solution, notice that po is the atmospheric pressure at sea level; for if h = 0, p = Poe-mo A more convenient form of the given equation, for purposes of computation, is obtained by equating the common logarithms of both sides of the equation, which gives: These equations may be solved simultaneously for the constants m and Po, as follows: 2. Each of the persons receiving one of the letters mentioned in Exercise 18, page 232 was requested to forward 10 cents. Find the number on the letters at which the chain should have been completed if the object had been to raise $1,000,000. 3. The population of the United States in 1898 was approximately 75,000,000. Find the number on the letter at which the chain in Exercise 18, page 232, should have been completed if it was expected that every man, woman, and child in the country would receive one. 4. Would the method of Section 85 enable one to solve the equation 2*+ 3* = 7? = 5. Find a function of the type y xn which passes through the point (3,2). Find the average rate of change of this function from x = 3 to x = 4. 6. Find the common logarithm of any positive number x, assuming that a table of logarithms to the base e is given. = 7. Find the numerical value of M = log10 € (e 2.718) from the table of common logarithms. By the method of Example 2, Section 85, find the value of m = loge 10. Find the reciprocal of M, and compare it with m. 8. Assuming that a table of logarithms to the base b is given, show that the logarithm of any positive number x to the base a is Illustrate these two equations on the graphs of log. x and log10 X. NOTE: The system of common logarithms, in which the base is 10, and that of natural logarithms, in which the base is e = 2.718, are the most important systems of logarithms. The former is used for numerical computations, the latter for most theoretical work. The relations be tween these two systems obtained in Exercises 6 to 9 may be summed up as follows: If the common logarithm of a number is given, the natural logarithm is found by multiplying it by The constant m is called the modulus for changing from the common to the natural system. If the natural logarithm of a number is given, the common logarithm is found by multiplying it by which is called the modulus for changing from the natural to the common system. 10. Solve Example 3, Section 85, using the table of natural logarithms (page 31 of the Tables). 11. Find log2 3, log2 5, log2 7, and then by means of some of the fundamental properties of logarithms (Section 81, properties 7, 8, 9) construct a table of logarithms to the base 2 for the integers 1 to 10 inclusive. 12. A hawser of a ship is subjected to a strain of 6 tons. How many turns must be taken around a post in order that a man who can not pull more than 200 pounds may keep the hawser from slipping, if the coefficient of friction m = 0.175 and S strain in the hawser in the angle of contact of the == Peme, where pull of the man in pounds, ✪ hawser and post in radians. pounds, P = = S == 13. A steamer approaching a dock has a velocity of 20 feet per second at the instant the power is shut off, and 10 feet per second at the end of 1 minute. Find the velocity at the end of 2 minutes if the law of motion is v = Voe-mt 14. In a chemical experiment it was found that at the end of 1 hour there were 35.6 c.c. of a substance in solution and at the end of 3 hours, 18.5 c.c. If the amount A of the substance at any time t is given by the equation A = ke-mt, determine the time that elapsed before the amount was reduced to 5 c.c. 15. The intensity of light I after passing through a medium of thickness T is I = I。e-mT. If light loses 3% of its intensity in passing through a lens, what per cent of intensity will remain after passing through four lenses? 16. Radium decomposes so that the amount A remaining after a time t is A Aoe-mt. If % disappears in 20 years, how long before one-half of the original amount will be gone? = |