But if, instead of the logarithm of the first term, we substi tute its arithmetical complement, this may be added to the sum of the other two, or more simply, all three may be added together, by one operation. After the index is diminished by 10, the result will be the same as by the common method. For subtracting a number is the same, as adding its arithmetical complement, and then rejecting 10, 100, or 1000, from the sum. (Art. 53.)

In the following proportion, the calculation is made in both ways.

If the profit on 2625 dollars employed in trade, is 831 dollars; what is the profit on 536 dollars?

The second method here, after rejecting 10, gives the same result as the first. But it is unnecessary, first to add two of the terms, and then the arithmetical complement of the other. The three may be added at once; and it will generally be expedient, to place the terms in the same order, in which they are arranged in the statement of the proportion.

57. In compound, as in single proportion, the term required may be found by logarithms, if we substitute addition for multiplication, and subtraction for division.

Ex. 1. If the interest of $365, for 3 years and 9 months, be $82.13; what will be the interest of $8940, for 2 years and 6 months?

In common arithmetic, the statement of the question is made in this manner,

365 dollars
3.75 years

: 82.13 dollars :: {8940 dollars

2.5 years

And the method of calculation is, to divide the product of the third, fourth, and fifth terms, by the product of the two first.* This, if logarithms are used, will be to subtract the sum of the logarithms of the two first terms, from the sum of the logarithms of the other three.

Sum of the logs. of the 3d, 4th, and 5th 6.26378

58. The calculation will be more simple, if, instead of subtracting the logarithms of the two first terms, we add their arithmetical complements. But it must be observed, that each arithmetical complement increases the index of the logarithm by 10. If the arithmetical complement be introduced into two of the terms, the index of the sum of the logarithms will be 20 too great; if it be in three terms, the index will be 30 too great, &c.

The result is the same as before, except that the index of the logarithm is 20 too great.

Ex. 2. If the wages of 53 men for 42 days be 2200 dollars; what will be the wages of 87 men for 34 days?

59. In the same manner, if the product of any number of quantities, is to be divided, by the product of several others; we may add together the logarithms of the quantities to be divided, and the arithmetical complements of the logarithms of the divisors.

Ex. If 29.67 x 346.2 be divided by 69.24 x 7.862 × 497; what will be the quotient?

In this way, the calculations in Conjoined Proportion may *be expeditiously performed.

60. In calculating compound interest, the amount for the first year, is made the principal for the second year; the amount for the second year, the principal for the third year, &c. Now the amount at the end of each year, must be portioned to the principal at the beginning of the year. If the principal for the first year be 1 dollar, and if the amount of 1 dollar for 1 year =a; then, (Alg. 377.)

pro

a a2 the am't for the 2d y'r, or the prin. for the 3d; 1:a: a:a3the am't for the 3d y'r, or the prin. for the 4th; a3: a1=the am't for the 4th y'r, or the prin. for the 5th,

3

[&c.

That is, the amount of 1 dollar for any number of years is obtained, by finding the amount for 1 year, and involving this to a power whose index is equal to the number of years. And the amount of any other principal, for the given time, is found, by multiplying the amount of 1 dollar, into the number of dollars, or the fractional part of a dollar.

If logarithms are used, the multiplication required here may be performed by addition; and the involution, by multiplication. (Art. 45.) Hence,

61. To calculate Compound Interest, Find the amount of 1 dollar for 1 year; multiply its logarithm by the number of years; and to the product, add the logarithm of the principal. The sum will be the logarithm of the amount for the given time. From the amount subtract the principal, and the remainder will be the interest.

If the interest becomes due half-yearly or quarterly; find the amount of one dollar, for the half-year or quarter, and

F

multiply the logarithm, by the number of half-years or quarters in the given time.

Ex. 1. What is the amount of 20 dollars, at 6 per cent compound interest, for 100 years?

2. What is the amount of 1 cent, at 6 per cent compound

More exact answers may be obtained, by using logarithms of a greater number of decimal places.

3. What is the amount of 1000 dollars, at 6 per cent comAns. 1790.80. pound interest, for 10 years?

EXPONENTIAL EQUATIONS.

62. An EXPONENTIAL equation is one in which the letter expressing the unknown quantity is an exponent.

Thus a=b, and a=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?