2.0 Properties of a logarithmic functions

  Logarithms and Properties

1. ${\log _a}1 = 0$
2. ${\log _a}a = 1$
3. ${\log _a}xy = {\log _a}x + {\log _a}y$
4. ${\log _a}\left( {\frac{x}{y}} \right) = {\log _a}x - {\log _a}y$
5. ${\log _a}{x^n} = n{\log _a}x$
6. ${\log _a}{x^{2k}} = 2k{\log _a}\left| x \right|$
7. ${\log _{{a^m}}}x = \frac{1}{m}{\log _a}x$
8. ${\log _{{a^{2k}}}}x = \frac{1}{{2k}}{\log _{\left| a \right|}}x$
9. ${\log _{{a^n}}}{b^m} = \frac{m}{n}{\log _a}b$
10. ${\log _a}b = \frac{{{{\log }_c}b}}{{{{\log }_c}a}}$
11. ${\log _a}b = \frac{1}{{{{\log }_b}a}}$
12. ${\log _a}b \times {\log _b}c \times {\log _c}a = 1$
13. ${a^{{{\log }_a}x}} = x$
14. ${a^{{{\log }_b}x}} = {x^{{{\log }_b}a}}$