1. Logaritham :
If $a$ is a positive real number, other than $1$ and $a^{m}$ $=X$, then $m=log_{a}X$.
e.g.
i) $2^{3}$ $=8$ $\Rightarrow log_{2}8$ $=3$
ii)$2^{-3}$ $=\frac{1}{8}$ $\Rightarrow log_{2}
frac{1}{8}$ $=-3$
iii) $(.1)^{2}$ $=.01$ $\Rightarrow log_(.1).01$ $=2$.
2. Properties of Logarithms :
i). $log_{a}{(XY)}$ $=log_{a}X$ $+log_{a}Y$
ii). $log_{a}\left(\frac{X}{Y}\right)$ $=log_{a}X$ $-log_{a}Y$
iii). $log_{x}x$ $=1$.
iv). $log_{a}1$ $=0$
v). $log_{a}(X^{n})$ $=n(log_{a}X)$
vi). $log_{a}X$ $=\frac{1}{log_{X}a}$
vii). $log_{a}X$ $=\frac{log_b{X}}{log_{b}a}$ $=\frac{logX}{loga}$.
3. Common Logarithms :
A logaritham to the base 10 is known as common logarithms.
e.g. log 100 = 2 since $10^{2}$ $=100$.
4. The integral part of a common logarithm is known as the characteristic and the non-negative decimal part is called the mantissa.
e.g. log $39.2$ $=1.5933$, then $1$ is the characteristic and $5933$ is the mantissa of the logarithm.
To find Characteristic and Mantissa
Characteristic is determined by inspection and the mantissa by logarithmic table.
Case I, to find the characteristic of the logarithm of a number greater than 1:
The characteristic is one less than the number of digits in the left of the decimal point in the given number.
e.g. 654.24 , characteristic = 2
Case II, to find the characteristic of the logarithm of a number less than 1:
The characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and it is negative.
e.g. 0.6453, characteristic = $\overline{1}$
0.06134, characteristic = $\overline{2}$.
