204 AlvGKBRA. 



16. CoROLiyARY. Moving the decimal point in a numbei' one 

 place to the right increases its coin^non logarithm by i , ayid moving 

 it one place to the left decreases its logarithm by i. 



17. C0R01.LARY. The commo7i logarithms of all numbers con: 

 sisti7ig of the same significayit figures have the same, mantissa. 



For moving the decimal point merely adds or subtracts i from 

 the logarithm ; that is, merely affects the characteristic. Thus 

 log 256=2.40824 

 log 25.6=1.40824 

 log 2.56=0.40824 

 log .256=1.40824 

 log .0256=2.40824 

 log .00256=3.40824 



18. ThkorKM. If a number has its first significant figure in 

 units' place, the characteristic of its common logarith?n is o. 



If the number has its first significant figure in units' place, the 

 value of the number must lie somewhere between i and 10. But 

 the logarithm of i is o and the logarithm of 10 is i. Hence the 

 logarithm of the proposed number must lie somewhere between o 

 and I. (Art. 3.) That is, its characteristic must be o. 



Thus log 2.56=0.4082400 



and log 9-99=o-9995655 



19. ThkorBM. The characteristic of the commoji logarithm of a 

 number cguals the nu?nber of places the first significant figure of the 

 number is removed fro7n miits' place, arid is positive if the first sig- 

 nificant fig uj^e stands to the left of units' place and is negative if it 

 stands to the right of units' place. 



By the previous article, if the first significant figure stands in 

 units place the characteristic's o. If the first significant figure 

 stands in the ^th place to the left of units place, then the char- 

 acteristic of its logarithm must be a number such that it can be made 

 from o by adding i to it n times. (iVrt. 15.) In other words, the 

 characteristic must be 71. 



If the first significant figure of the given number stands in the 

 nth place to the right of units place, then the characteristic of its 



