Logarithm Formulas

1.  y = {\log  _a}x if and only if x = {a^y}, x > 0 and y \in \mathbb{R}, ais positive number.

2.  {\log  _a}1 = 0

3.  {\log  _a}a = 1, {\log _e}e = 1, i.e. \ln e = 1

4.  {\log  _a}mn = {\log _a}m + {\log _a}n

5.  {\log  _a}\left( {\frac{m}{n}} \right) = {\log _a}m - {\log _a}n

6.  {\log  _a}{(m)^n} = n{\log _a}m

7.  {\log  _a}m = {\log _b}m \cdot {\log _a}b

8.  {\log  _b}m = \frac{{{{\log }_a}m}}{{{{\log }_a}b}}

9.  {\log  _b}a = \frac{1}{{{{\log }_a}b}}

10. {\log _a}x = \frac{{\ln x}}{{\ln a}} = ({\log  _a}e)\ln x

11. \log \frac{{bc}}{{{a^2}}} + \log  \frac{{ac}}{{{b^2}}} + \log \frac{{ab}}{{{c^2}}} = 0

12. \log n! = \log 2 + \log 3 + \log 4 + \cdots  + \log n