3. Find the fifth power of .87451, by logarithms. Log of .87451 1.9417648 5 Where 5 times the negative index 1, being - 5, and +4 to carry, the index of the power is 1. 4. Find the 365th power of 1.0045, by logarithms. Log. 1.0045 0.0019499 365 . 97495 116994 58497 5. Required the square of 6.05987, by logarithms. Ahs. 36.72203 6. Required the cube of .176546, by logarithms. Ans. .005502674 .7. Required the 4th power of .076543, by logarithms. Ans. 0000343259 8. Required the 5th power of 2.97643 by logarithms. Ans. 233.6031 9. Required the 6th power of 21.0576 by logarithms. Ans. 87187340 10. Required the 7th power of 1.09684, by logarithms. Ans. 1.909864 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 both of 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 ren, der 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. EXAMPLES. 1. Find the square root of 27.465, by logarithms. Log. of 27.465 2)1.4387796 2. Find the cube root of 35.6415, by logarithms. Log. of 35.6415 . 3)1.5519560 Root 3.29093 .5173186 3. Find the 5th root of 7.0825, by logarithms. Log. of 7.0825 5)0.8501866 4. Find the 365th root of 1.045, by logarithms. Log, of 1.045 365)0.0191163 Root 1.000121 0.0000524 5. Find the value of 0.001234) by logarithms.. Log. of 001234 3.0913152 2 3)6.1826304 Ans. .00115047 2.0608768 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. Find the value of (.024554) by logarithms. Log. of .024554 2.3901223 3 2)5.1703669 Ans. 00384754 3.5851834 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. Ans. •9918624 10. Required the 7th root of .098674, by logarithms. Ans. .7183146 21 373 Ans. .146895 112 Ans.-.1937115 11. Required the value of (373) *, by logarithms. 1 2. Required the cube root of by logarithms. 3.14159' Ans. .6827842 3. Required the .07 power of .00563, by logarithms. Ans. .6958821 x 4. Required the value of -, by logarithms. 17} Ans. .04279825 1 5 7 X.012 6. Required the value of #VIX.03 Y15, by lo rithms. Ans. 001165713 Š ; 7} V12} X.194/17 garithms. Ans. .3009158638 7. Required the value of 127 & 19+ 1V361 3), by 4 141-192837" logarithms. Ans. 49.38712 226 MISCELLANEOUS QUESTIONS. 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. Sh. 43min. 38 i 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, betted three shillings to two upon every deal, and afier twenty deals found he had gained five shillings ; how many deals did he win? Ans. 13. 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 B, running together, in twelve hours, and by the cock A alone in twenty hours ; in what time will it be filled by the cock B 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. b Ans. d (tot e'f ' g 8. What number is that, which being severally added to 3, 19, and 51, shall make the results in geometrical progression ? Ans. 13 |