Exponent and Logarithm Laws

$a^m \cdot a^n = a^{m+n}$, $\frac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$

$a^0 = 1$, $a^{-n} = \frac{1}{a^n}$, $a^{m/n} = \sqrt[n]{a^m}$

$\log_a(xy) = \log_a x + \log_a y$

$\log_a(\frac{x}{y}) = \log_a x - \log_a y$

$\log_a(x^n) = n\log_a x$

$\log_a a = 1$, $\log_a 1 = 0$

$\log_a x = \frac{\log_b x}{\log_b a}$ (change of base)

$\ln x = \log_e x$. $e^{\ln x} = x$. $\ln(e^x) = x$.

$3^x = 20$ → $x = \frac{\ln 20}{\ln 3} \approx 2.73$

$\ln(2x + 1) = 4$ → $2x + 1 = e^4$ → $x = \frac{e^4 - 1}{2} \approx 26.8$