Logarithms and exponents

See exponential function and logarithmic function.

Formulas

For $a^x=b$, then $x={\log }_{a}b$.

$${\log }_{a}xy={\log }_{a}x+{\log }_{a}y$$ $${\log }_a\frac{x}{y}={\log }_{a}x-{\log }_{a}y$$ $${\log }_a{x}^m=m{\log }_{a}x$$ $${\log }_{a}x=\frac{{\log }_{b}x}{{\log }_{b}a}$$ $$a^x=e^{x\ln a}$$

Complex case

A complex exponent will have infinitely many values, because of the nature of the complex-valued logarithmic function.

$$a^z=e^{z\ln a}$$