$$\log_{a}{1} = 0$$
$$\log_{a}{xy} = \log_{a}{x}+\log_{a}{y}$$
$$\log_{a}{b}^{c} = c \log_{a}{b}$$
$${b}^{\log_{b}{x}} = x$$
$${x}^{\log_{b}{y}} = {y}^{\log_{b}{x}}$$
$${\log_{a}{\sqrt[n]{b}}} = \dfrac{\log_{a}{b}}{n}$$
$${\log_{a}{b}} = \dfrac{\log_{c}{b}}{\log_{c}{a}}$$
$${\log_{a}{b}}*{\log_{b}{a}}= 1$$
$$\log_{b^n}{a}=\dfrac{1}{n}\log_ba\ $$
$$a^m\times\ a^n=a^{m+n}$$
$$\frac{a^m\ \ }{a^n}\ =a^{m-n}$$
$$\left(a^m\right)^{^n}=a^{m\times\ n}$$
$$\left(a\times\ b\right)^m\ =a^m\times\ b^m$$
$$a^{-m}=\ \frac{1}{a^m}$$
$$a^{\frac{m}{n}}=\sqrt[\ n]{a^m}$$
$$log_a(x/y) = log_a(x) − log_a(y)$$
$$log_a(a) = 1$$
$$log_a(x) = 1/log_x(a)$$
Number of digits formula: $$⌊log₁₀(N)⌋ + 1$$
- Logarithms can be used to quickly find the number of digits in an exponent.
