xxxiv LOCAIMTIIMS. 



Powers and Roots of Numbers greater than Unity. 



Examples. Desired the cube of 47. 16, i.e. (47. i6) 3 . 



log 47. 1 6 =1-6735 



Multiply by 3 



log x = 5.0205 .. x = ix^S-io 6 , or 104800. 



Desired ^471.6. 



log 47 1. 6 =2.6735 



Divide by 5 



logs =0.5347 .'.x= 3.425. 



Example. Desired the 2.416 power of 65 830. 



log 65 830. = 4.8184 



Multiply by 2.416 



96368 

 i 92736 

 4818 

 2892 

 .-. log (65 830)2.410 = 11.6413 



(65 830)2-416 = 4.378-1011 = 437 800 ooo ooo. 



To avoid the direct multiplication of the logarithm by the index of power, 

 when this contains several figures, their logs may be taken. Thus 



(5 pi. logs.) (4 i>l. logs.) 



log log 65 830 = log 4.8184 = 0.68 291 0.6829 



log 2.416 = 0.38310 0.3831 



log log (65 830)2-416 = i. 06 601 i. 0660 



antilog i. 06 601 = log (65 830)2-416 = 1 1.641 1 1.64 



* (65830)2-416= 4.376-1011 4.365-1011. 



When the log of a log is thus taken, a table giving at least one 

 place more in the mantissa should be used than would be otherwise 

 needed in order to protect the last place of figures in the result, as 

 is shown in the above example, which is by no means an extreme 

 case. If the quantity is a decimal fraction, the negative character- 

 istic must be separately treated and the result subtracted from the 

 result obtained with the mantissa. 



Powers of Decimal Fractions. Here it is necessary to notice that 

 the logarithm of a decimal fraction is composed of two parts, one 

 positive the other negative. Thus, 



log 0.06 831 =1.8345 or 8.8345 10., 



according to the notation chosen. In the first form (the one here 

 recommended) the logarithm consists of a negative characteristic 



