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EVOLUTION, OR THE EXTRACTION OF ROOTS, BY LOGARITHMS.
TAKE out the logarithm of the given number from the table, and divide it by 2 for the square root, 3 for the cube root, &c., and the natural number, answering to the result, will be the root required.
But if it be a compound root, or one that consists of both a root and a power, multiply the logarithm of the given number by the numerator of the index, and divide the product by the denominator, for the logarithm of the root sought.
Observing, in either case, when the index of the logarithm is negative, and cannot be divided without a remainder, to increase it by such a number as will render it exactly divisible; and then carry the units borrowed, as so many tens, to the first figure of the decimal part, and divide the whole accordingly.
1. Find the square root of 27.465, by logarithms. Log. of 27.465
2. Find the cube root of 35.6415, by logarithms.
Root 3.29093 .
3. Find the 5th root of 7.0825, by logarithms. Log. of 7.0825
4. Find the 365th root of 1.045, by logarithms.
5. Find the value of (.001234), by logarithms.
Here, the divisor 3 being contained exactly twice in the negative index 6, the index of the quotient, to be put down,
will be 2.
6. Find the value of (.024554)3, by logarithms.
Log. of .024554
11. Required the value of
12. Required the value of
Here 2 not being contained exactly in 5, 1 is added to it, which gives 3 for the quotient; and the 1 that is borrowed, being carried to the next figure, makes 11, which, divided by 2, gives .58, &c.
7. Required the square root of 365.5674, by logarithms. Ans. 19.11981.
8. Required the cube root of 2.987635, by logarithms. Ans. 1.440265.
9. Required the 4th root of .967845, by logarithms.
10. Required the 7th root of .098674, by logarithms. Ans. .7183146.
2. Required the cube root of
(373), by logarithms.
(112), by logarithms.
MISCELLANEOUS EXAMPLES IN LOGARITHMS.
1. Required the square root of
3. Required the .07 power of .00563, by logarithms.
4. Required the value of
7. Required the value of
(#)* × (3)3
5. Required the value of x.012 by loga11'
6. Required the value of
111 X.03 15}
713/12 × .194/171
1. A person being asked what o'clock it was, replied that it was between eight and nine, and that the hour and minute hands were exactly together; what was the time? Ans. 8h. 43 min. 38 sec. 2. A certain number, consisting of two places of figures, is equal to the difference of the squares of its digits, and if 36 be added to it, the digits will be inverted; what is the number? Ans. 48.
3. What two numbers are those, whose difference, sum, and product, are to each other as the numbers 2, 3, and 5, respectively? Ans. 2 and 10.
4. A person, in a party at cards, bet three shillings to two upon every deal, and after twenty deals found he had gained five shillings; how many deals did he win?
5. A person wishing to enclose a piece of ground with palisades, found, if he set them a foot asunder, that he should have too few by 150, but if he set them a yard asunder he should have too many by 70; how many had he? Ans. 180.
6. A cistern will be filled by two cocks, a and в, running
together, in twelve hours, and by the cock a alone in twenty
hours; in what time will it be filled by the cock в alone? Ans. 30 hours.
7. If three agents, A, B, C, can produce the effects, a, b, c, in the times e, f, g, respectively; in what time would they jointly produce the effect d.
8. What number is that, which being severally added to 3, 19, and 51, shall make the results in geometrical progression? Ans. 13.
9. It is required to find two geometrical mean proportionals between 3 and 24, and four geometrical means between 3 and 96. Ans. 6 and 12; and 6, 12, 24, and 48. 10. It is required to find six numbers in geometrical progression such, that their sum shall be 315, and the sum of the two extremes 165. Ans. 5, 10, 20, 40, 80, and 160.
11. The sum of two numbers is a, and the sum of their
reciprocals is b, required the numbers.
12. After a certain number of men had been employed on a piece of work for 24 days, and had half finished it, 16 men more were set on, by which the remaining half was completed in 16 days; how many men were employed at first; and what was the whole expense, at 1s. 6d. a day per man? Ans. 32 the number of men; and the whole expense 1157. 4s.
13. It is required to find two numbers such, that if the squares of the first be added to the second, the sum shall be 62, and if the square of the second be added to the first, it shall be 176. Ans. 7 and 13. 14. The fore wheel of a carriage makes six revolutions more than the hind wheel, in going 120 yards; but if the circumference of each wheel was increased by three feet, it would make only four revolutions more than the hind wheel in the same space; what is the circumference of each wheel? Ans 12 and 15 feet.
15. It is required to divide a given number a into two such parts, x and y, that the sum of me and ny shall be equal to some other given number b.
16. Out of a pipe of wine, containing 84 gallons, 10 gallons were drawn off, and the vessel replenished with 10 gallons of water; after which 10 gallons of the mixture were again drawn off, and then 10 gallons more of water poured in; and so on for a third and fourth time; which being done, it is required to find how much pure wine remained in the vessel, supposing the two fluids to have been thoroughly mixed each time? Ans. 484 gallons. 17. A sum of money is to be divided equally among a
certain number of persons; now, if there had been 3 claimants less, each would have had 1501. more, and if there had been 6 more, each would have had 1207. less; required the number of persons and the sum divided. Ans. 9 persons; sum 27007. 18. From each of 16 pieces of gold, a person filed the worth of half a crown, and then offered them in payment for their original value, but the fraud being detected, and the pieces weighed, they were found to be worth in the whole, no more than eight guineas; what was the original value of each piece?
19. A composition of tin and copper, containing 100 cubic inches, was found to weigh 505 ounces; how many ounces of each did it contain, supposing the weight of a cubic inch of copper to be 51 ounces, and that of a cubic inch of tin 44 Ans. 420 oz. of copper, and 85 oz. of tin, 20. A privateer, running at the rate of 10 miles an hour, discovers a vessel 18 miles ahead of her, making way at the rate of 8 miles an hour; how many miles will the latter run before she is overtaken? Ans. 72 miles. 21. In how many different ways is it possible to pay 1007. with seven shilling pieces, and dollars of 4s. 6d. each?
Ans. 31 different ways. 22. Given the sum of two numbers 2, and the sum of their ninth powers 32, to find the numbers by a quadratic equation. Ans. 1±√(6 √✓ 34 — 33). 23. Given y3 — xy 666, and 3xy 406, to find x and y. Ans. = x 7, and y = 9. 24. The arithmetical mean of two numbers exceeds the geometrical mean by 13, and the geometrical mean exceeds the harmonical mean by 12; what are the numbers?
Ans. 234 and 104.
25. Given x3y + y3x = 3, and x ya ¦ y°x3· 7, to find the values of x and y. Ans. ∞ = 1 x · 2 ( √5 + 1), y = 1 ( √5 26. Given x + y + z=23, xy + xz + yz = 167, and xyz 385, to find x, y, and z. Ans. x= 5, y = 7, z = 11. 27. To find four numbers, x, y, z, and w, having the product of every three of them given; viz. xyz = 231, xyw 420, yzw 1540, and xzw = 660. Ans. x= 3, y=17, z= 17, z = 11, and w 20. 28. Given x+yz 384, y+xz=237, and z + xy = 192, to find the values of x, y, and z.
Ans. x= 10, y 7, and z 29. Given x2 + xy — 108, y2+yz≈ 69, and x2 +xz=580, to find the values of x, y, and z.
9, y = 3, and z