Logarithms

Positive real numbers : x, y, a, c, k

Natural number : n

$$ 1.\ \log_a1=0 $$

$$ 2.\ \log_aa=1 $$

$$ 3.\ \log_a0=-\infty \ \text{if} \ a \gt 1\ \text{and} \ +\infty\ \text{if} \ a \lt 1 $$

$$ 4.\ \log_a(xy)= \log_ax+\log_ay $$

$$ 5.\ \log_a\left( \frac xy \right)= \log_ax-\log_ay $$

$$ 6.\ \log_a(x^n)=n \log_ax $$

$$ 7.\ \log_a\sqrt [n]{x} = \frac 1n \log_ax $$

$$ 8.\ \log_ax=\frac {\log_cx}{\log_ca}=\log_cx-\log_ac , \ c \gt 0, c\neq 1 $$

$$ 9.\ \log_ac= \frac {1}{\log_ca} $$

$$ 10.\ x=a^{\log_ax} $$

$$ 11.\ \log_{10}x=\log x $$

$$ 12.\ \log_ex=\ln x $$

$$ 13.\ \log_x=\frac {1}{\ln 10} 1n x= 0.434294 \ 1n x $$

$$ 14.\ \ln x= \frac {1}{\log_e} \log x = 2.302585 \log x $$