natural logarithm

Properties of Logarithm

1. If   $y = a^x$,   then   $\log_a y = x$.   ← Definition of logarithm
2. $\log_a xy = \log_a x + \log_a y$
3. $\log_a \dfrac{x}{y} = \log_a x - \log_a y$
4. $\log_a x^n = n \log_a x$
5. $\log_a a = 1$
6. $\log_a 1 = 0$
7. $\log_{10} x = \log x$   ←   Common logarithm
8. $\log_e x = \ln x$   ←   Naperian or natural logarithm
9. $\log_y x = \dfrac{\log x}{\log y} = \dfrac{\ln x}{\ln y}$   ←   Change base rule
10. If   $\log_a x = \log_a y$,   then   $x = y$.
11. If   $\log_a x = y$,   then   $x = {\rm antilog}_a \, y$.