Plane trigonometry, by S.L. Loney.

150

TRIGONOMETRY.

(i) Let the number be greater than unity.
Since 10~ = 1, therefore log =0;
since 101 =10, therefore log10 =1;
since 10 = 100, therefore log 100 =2,
and so on.
Hence the logarithm of any number lying between 1
and 10 must lie between 0 and 1, that is, it will be a
decimal fraction and therefore have 0 as its characteristic.
So the logarithm of any number between 10 and 100
must lie between 1 and 2, i.e. it will have a characteristic
equal to 1.
Similarly the logarithm of any number between 100
and 1000 must lie between 2 and 3, i.e. it will have a
characteristic equal to 2.
So, if the number lie between 1000 and 10000, the
characteristic will be 3.
Generally, the characteristic of the logarithm of any
number will be one less than the number of digits in its
integral part.
Exs. The number 296-3457 has 3 figures in its integral part and
therefore the characteristic of its logarithm is 2.
The characteristic of the logarithm of 29634-57 will be 5-1, i.e. 4.
(ii) Let the number be less than unity.
Since 100 = 1, therefore log 1=0:
since 10-1= -I 1, therefore log 1 = -1;
10
since 10-2= 1- 01, therefore log -01 =- 2;
102
since 101-3 = -001, therefore log 001 = -3;
and so on.