ence be added to, or subtracted from, the difference between the ratio of the births and that of the deaths, the whole rate of increase will be obtained. Ex. 11. If in a country, the ratio of births be the ratio of immigration, and if the population this year be 10 millions, what will it be 20 years hence? The rate of the natural increase And the population at the end of 20 years, is 12,611,000. 12. If the ratio of the births be of emigration 509 in what time will three millions increase to four and a half millions? If the period in which the population will double be given; the numbers for several successive periods, will evidently be in a geometrical progression, of which the ratio is 2; and as the number of periods will be one less than the number of terms; Ex. 1. If the descendants of a single pair double once in 25 years, what will be their number, at the end of one thousand years? The number of periods here is 40. 2. If the descendants of Noah, beginning with his three sons and their wives, doubled once in 20 years for 300 years, what was their number, at the end of this time? Ans. 196,608. 3. The population of the United States in 1820 being 9,625,000; what must it be in the year 2020, supposing it to double once in 25 years? Ans. 2,464,000,000. 4. Supposing the descendants of the first human pair to double once in 50 years, for 1650 years, to the time of the deluge, what was the population of the world, at that time? EXPONENTIAL EQUATIONS. 62. An EXPONENTIAL equation is one in which the letter expressing the unknown quantity is an exponent. Thus, ab, and = bc, are exponential equations. These are most easily solved by logarithms. As the two members of an equation are equal, their logarithms must also be equal. If the logarithm of each side be taken, the equation may then be reduced, by the rules given in algebra. Ex. What is the value of x in the equation 3=243? Ꮖ Taking the logarithms of both sides, log. 3*= log. 243. But the logarithm of a power is equal to the logarithm of the root, multiplied into the index of the power. (Art. 45.) Therefore (log. 3)×x=log. 243; and dividing by log. 3. log. 243 2.38561 x= 5. So that 35 = 243. 63. The preceding is an exponential equation of the simplest form. Other cases, after the logarithm of each side is taken, may be solved by Trial and Error, in the same manner as affected equations. (Alg. 503.) For this purpose, make two suppositions of the value of the unknown quantity, and find their errors; then say, As the difference of the errors, to the dif- So is the least error, to the correction required = Ex. 1. Find the value of x in the equation — 256. Taking the logarithms of both sides (log. x) xx=log. 256. Let x be supposed equal to 3.5, or 3.6. Then, 0.09844 0.1 0.40556 0.4119, the correction. This added to 3.6, the second assumed number, makes the value of x-4.0119. To correct this farther, suppose x=4.011, or 4.012. By the first supposition. By the second supposition. x=4.011, and log. x=0.60325 x=4.012, and log. x= =0.60336 Multiplying by 4.012 Then, 0.00105 0.001 0.01139 0.011 very nearly. Subtracting this correction from the first assumed number 4.011, we have the value of x=4, which satisfies the conditions of the proposed equation; for 41=256. 2. Reduce the equation 4x-100x3. 3. Reduce the equation -9x. Ans. x=5. 64. The exponent of a power may be itself a power, as in the equation where x is the exponent of the power m2, which is the expo amt nent of the power am. x Ex. 4. Find the value of x, in the equation 93 -1000. 3x (log. 9)= log. 1000. Therefore, 3*= Then, as 3*=3.14. (log. 3) = log. 3.14. Therefore, x= log. 1000 = = 3.14. In cases like this, where the factors, divisors, &c., are logarithms, the calculation may be facilitated, by taking the logarithms of the logarithms. Thus, the value of the fraction 4 is most easily found, by subtracting the logarithm. of the logarithm which constitutes the denominator, from the logarithm of that which forms the numerator. ba+d 5. Find the value of x, in the equation- =m. с SECTION IV. DIFFERENT SYSTEMS OF LOGARITHMS, AND COMPUTATION OF THE TABLES. 65. For the common purposes of numerical computation, Briggs' system of logarithms has a decided advantage over every other. But the theory of logarithms is an important instrument of investigation, in the higher departments of mathematical science. In its numerous applications, there is frequent occasion to compare the common system with others; especially with that which was adopted by the celebrated inventor of logarithms, Lord Napier. In conducting these investigations, it is often expedient to express the logarithm of a number, in the form of a scries. If a N, then x is the logarithm of N. (Art. 2.) = To find the value of x, in a series, let the quantities a and N be put into the form of a binomial, by making a=1+b, and N=1+n. Then (1+6)=1+n, and extracting the root y of both sides, we have Ꮖ As these expressions will be the same, whatever be the 1 value of y, let y be taken indefinitely great; then - and being indefinitely small, in comparison with the numbers —I, -2, &c., with which they are connected, may be cancelled Ꮖ y y from the factors (1)(2). &c. (− 1 ) (2) -2 (Alg. 456.) leaving 1+6) + ( 3 ) (+), &c 2 &c. |