Logarithms, Surds and Indices

$$\log_{a}{1} = 0$$

$$\log_{a}{xy} = \log_{a}{x}+\log_{a}{y}$$

$$\log_{a}{b}^{c} = c \log_{a}{b}$$

$$\log_{b^n}{a}=\dfrac{1}{n}\log_ba\ $$

$$log_a(x/y) = log_a(x) − log_a(y)$$ 

$$log_a(a) = 1$$

• Rationalization: 1/(√a + √b) = (√a − √b)/(a−b)

• $$ \frac{1}{\sqrt{N+1}-\sqrt{N}} ={\sqrt{N+1}+\sqrt{N}}$$

$${\log_{a}{b}} = \dfrac{\log_{c}{b}}{\log_{c}{a}}$$

$${\log_{a}{b}}*{\log_{b}{a}}= 1$

$$log_a(x) = 1/log_x(a)$$

$$a^m\times\ a^n=a^{m+n}$$

$$\frac{a^m\ \ }{a^n}\ =a^{m-n}$$

$$\left(a^m\right)^{^n}=a^{m\times\ n}$$

$$\left(a\times\ b\right)^m\ =a^m\times\ b^m$$

$$a^{-m}=\ \frac{1}{a^m}$$

$$a^{\frac{m}{n}}=\sqrt[\ n]{a^m}$$

If $$a > 1: log_a(x) > log_a(y) ⟺ x > y$$

If $$0 < a < 1: log_a(x) > log_a(y) ⟺ x < y $$

$${b}^{\log_{b}{x}} = x$$

$${x}^{\log_{b}{y}} = {y}^{\log_{b}{x}}$$

Number of digits formula: $$⌊log₁₀(N)⌋ + 1$$

• Logarithms can be used to quickly find the number of digits in an exponent.