Logarithm : If a is a positive real number, other than 1 and $ a^m=x $, then we write : $m=\log_{a}x $ and we say that the value of $\log x$ to the base $a$ is $m$.

Example 1 : $10^3=1000$ $\Rightarrow \log_{10}1000$ $=3 $

Example 2 : $10^{-2}=\cfrac{1}{100}$ $\Rightarrow \log_{10}{\cfrac{1}{100}}$ $= -2$

1. $ \log_{a}(xy)=\log_{a}x + \log_{a}y $ (Where $x$ , $y$ and a are a positive real numbers and $a$≠ 1)
2. $ \log_{a} \left(\cfrac{x}{y}\right)=\log_{a}x - \log_{a}y $ (Where $x$, $y$ and $a$ are a positive real numbers and $a$≠ 1)
3. $ \log_{x}{x}=1$
4. $ \log_{a}{1}=0$
5. $ \log_{a}{(x)^m}=m(\log_{a}x) $ (Where $a$ and $x$ are +ve real numbers, a ≠ 1 and $m$ is any real number.)
6. $ \log_{a}{x}=\cfrac{1}{log_{x}{a}}$(Where $a$ and $x$ are +ve real numbers, $a$ ≠ 1 and $x$ ≠ 1 )
7. $ \log_{a}{x} = \cfrac{\log_{b}{x}}{\log_{a}{b}} = \cfrac{\log{x}}{\log{a}} $

1. If base = 10, then we can write log a instead of $\log_{10}a$. $\log a$ is called as Common Logarithm of $a$.
2. If base = e, then we can write ln a instead of $\log_{e}a$. $\log a$ is called as Natural Logarithm of $a$.

Note:

1. e is a mathematical constant which is the base of the Natural Lograthim. It is known as Euler's number. It is also known as Napier's Constant.
2. $e= 1+ \cfrac{1}{1}$ $+\cfrac{1}{1 \cdot 2}+\cfrac{1}{1 \cdot 2 \cdot 3}$ $+\cfrac{1}{1 \cdot 2 \cdot 3 \cdot4}$ $+\cdot\cdot\cdot \approx2.71828$ $\\$

   In General:

   $e= 1+x+ $ $\cfrac{x^2}{2!}$ $+\cfrac{x^3}{3!}+\cfrac{x^4}{3!}+$ $\cdot\cdot\cdot$