Logarithm Formulas

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

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