93. 1. Given log 2.3010; find log 5. Given log 2.3010, log 3.4771, log 7 = .8451; find the values of the following: 94. In any system, the logarithm of any power of a quan tity is equal to the logarithm of the quantity multiplied by the exponent of the power. Assume the equation ax= =m; whence, x=logą m. Raising both members to the pth power, we have apm2; whence, loga m2 = px = p log, m. = 95. In any system, the logarithm of any root of a quantity is equal to the logarithm of the quantity divided by the index of the root. For, 1 1 log. Vm = log. (m2)=log. m (Art. 94). r 96. 1. Given log 2 = .3010; find the logarithm of 2§. Note. To multiply a logarithm by a fraction, multiply first by the numerator, and divide the result by the denominator. 2. Given log 3.4771; find the logarithm of V3. Given log 2.3010, log 3.4771, log 7.8451; find the By Art. 90, log (2 × 3)= log 2}+log 3 97. In the common system, the mantissa of the logarithms of numbers having the same sequence of figures are equal. To illustrate, suppose that log 3.053= .4847. log 30.53 = log (10 x 3.053) = log 10+ log 3.053 × log 305.3 = log (100 x 3.053)= log 100+ log 3.053 log .03053 = log (.01 × 3.053) = log .01 + log 3.053 = 810.4847 = 8.4847-10; etc. It is evident from the foregoing that if a number is multiplied or divided by any integral power of 10, thus producing another number with the same sequence of figures, the mantissæ of their logarithms will be equal. Thus, if log 3.053.4847, then Note. The reason will now be seen for the statement made in Art. 86, that only the mantissæ are given in a table of logarithms of numbers. For, to find the logarithm of any number, we have only to take from the table the mantissa corresponding to its sequence of figures, and the characteristic may then be prefixed in accordance with the rules of Art. 86. This property of logarithms is only enjoyed by the common system, and constitutes its superiority over others for the purposes of numerical computation. 98. 1. Given log 2.3010, log 3=.4771; find log .00432. log 432 = log (21 × 33) = 4 log 2+3 log 3 Then by Art. 97, the mantissa of the result is .6353. Whence by Art. 86, log .00432 7.6353 – 10. = EXAMPLES. Given log 2.3010, log 3.4771, log 7.8451; find the values of the following: By aid of this relation, if the logarithm of a quantity m to a certain base a is known, its logarithm to any other base b may be found by dividing by the logarithm of b to the base a. 100. To prove the relation log, ax log, b=1. Putting ma in the result of Art. 99, we have 101. The value of an arithmetical quantity, in which the operations indicated involve only multiplication, division, involution, or evolution, may be most conveniently found by logarithms. The utility of the process consists in the fact that addition takes the place of multiplication, subtraction of division, multiplication of involution, and division of evolution. In operations with negative characteristics the rules of Algebra must be followed. 102. 1. Find the value of .0631 x 7.208 × .51272. log (.0631 × 7.208 × .51272) By Art. 90, Adding, .*. log of result = 19.3677 - 20 = 9.3677 10 (see Note 1) Number corresponding to 9.3677 −10 = .23317. 2 Note 1. If the sum is a negative logarithm, it should be reduced so that the negative portion of the characteristic may be - 10. Thus, 19.3677-20 is reduced to 9.3677-10. Number corresponding = .04218. Note 2. To subtract a greater logarithm from a less, or to subtract a negative logarithm from a positive, increase the characteristic of the minuend by 10, writing 10 after the mantissa to compensate. Thus, to subtract 3.9022 from 2.5273, write the minuend in the form 12.527310; subtracting 3.9022 from this, the result is 8.6251 — 10. 3. Find the value of (.07396)5. By Art. 94, log (.07396)5 = 5 x log .07396. log.073968.8690 — 10 5 44.345050 = 4.3450 10 (see Note 1) |