Fundamentals

Exponent Laws

\[ \begin{align} a^0 & = 1 \\ a^1 & = a \\ a^{-p} & = \frac{1}{a^p} \\ a^\frac{1}{p} & = \sqrt[p]{a} \\ a^\frac{q}{p} & = (\sqrt[p]{a})^q = \sqrt[p]{a^q} \\ a^m \times a^n & = a^{m+n} \\ a^m \div a^n & = a^{m-n} \\ (a^m)^n & = a^{mn} \\ (ab)^n & = a^n b^n \\ \left(\frac{a}{b}\right)^n & = \frac{a^n}{b^n} \\ (\sqrt[n]{a})^n & = a \\ \sqrt[n]{a^n} & = \begin{cases} |a| & \mathrm{if}\ n\ \mathrm{is\ even} \\ \ a & \mathrm{if}\ n\ \mathrm{is\ odd} \end{cases} \\ \sqrt[n]{a} \times \sqrt[n]{b} & = \sqrt[n]{ab} \\ \frac{ \sqrt[n]{a} }{ \sqrt[n]{b} } & = \sqrt[n]\frac{a}{b} \\ \sqrt[m]{\sqrt[n]{a}} & = \sqrt[mn]{a} \\ \end{align} \]

Logarithm Laws

\[ \begin{align} y & = a^x \Longleftrightarrow x = \log_a y \\ \log_a a & = 1 \\ \log_a 1 & = 0 \\ \log_a (mn) & = \log_a m + \log_a n \\ \log_a \left(\frac{m}{n}\right) & = \log_a m - \log_a n \\ \log_a m^n & = n \log_a m \\ \log_a a^k & = k \\ \lg x & = \log_{10} x \\ \ln_x & = \log_e x \\ \log_a b & = \frac{\log_c b}{\log_c a} \\ \end{align} \]

Partial Fractions