If \(a > 0,\,\,a \ne 1\), then

1. \({\log _a}\left( {m.n} \right) = {\log _a}m + {\log _a}n\), where \(m\) and  \(n\) are positive numbers.
2. \({\log _a}{m^n} = n{\log _a}m\), where \(m\) is a positive number.
3. \({\log _a}\left( {\frac{m}{n}} \right) = {\log _a}m - {\log _a}n\), where  \(m\) and \(n\) are positive numbers.
4. For any positive real numbers \(r\) and \(b\), \(b \ne 1,\;\,{\log _b}r = \frac{{{{\log }_a}r}}{{{{\log }_a}b}}\)
5. \({\log _a}b\; = \frac{1}{{{{\log }_b}a}},\;{\rm{where}}\;b > 0,\,\,b \ne 1\) and \(a > 0,\,\,a \ne 1\)
6. \({\log _{{a^k}}}{m^n} = \frac{n}{k}{\log _a}m\,\,\), where \(m\) is positive. If the base is more than 1, then on increasing the base, value of the logarithm decreases. For example, \({\log _{10}}1000 = 3\), but \({\log _{1000}}1000 = 1\).
7. \({a^{{{\log }_a}}}^x = x\), where \(x\) is a positive number.
8. \({a^{{{\log }_c}x}} = {x^{{{\log }_c}a}}\) for any \(x\) and \(c\) ,which are positive and not equal to 1. For example \({3^{{{\log }_4}5}} = {5^{{{\log }_4}3}}\)
9. \({\log _a}x = {\log _a}y \Rightarrow x = y\)
10. \({\log _a}x = {\log _b}x \Rightarrow a = b\) or \(x\) = 1
11. \({\log _a}x > {\log _a}y\)\( \Rightarrow \left\{ {\begin{array}{*{20}{c}}{x > y}&{{\rm{if}}\;a > 1}\\{x < y}&{{\rm{if}}\;0 < a < 1}\end{array}} \right.\)
12. \({\log _a}{\log _b}{\log _c}d = e \Rightarrow d = {c^{{b^{{a^e}}}}}\)

13. \({\log _e}(1 + x) = x - \frac{{{x^2}}}{2} + \frac{{{x^3}}}{3} - \frac{{{x^4}}}{4} + .....\infty \) and 

    \({\log _e}(1 - x) =  - x - \frac{{{x^2}}}{2} - \frac{{{x^3}}}{3} - \frac{{{x^4}}}{4} - .....\infty \)