HSC Extension 1 and 2 Mathematics/Exponential and logarithmic functions

Index laws

$a^x\times a^y=a^{(x+y)}$

$a^x÷a^y=a^{(x-y)}$

Derivative of exponential functions

$\frac{d}{dx}{e}^x=e^x$

Derivative of e^kx, k a constant

$\frac{d}{dx}{e}^{kx}=k{e}^{kx}$

$\int e^{kx}dx=\frac{1}{k}{e}^{kx}+c$

Derivative of log_ex

$\frac{d}{dx}lo{g}_{e}x=\frac{1}{x},x>0$

$\int \frac{1}{x}dx=lo{g}_{e}x+c$

Derivative of log_e(ax), a > 0

$\frac{d}{dx}lo{g}_e(ax)=\frac{1}{x}$

Derivative of log_e(ax+b)

$\frac{d}{dx}lo{g}_e(ax+b)=\frac{a}{ax+b}$

$\frac{d}{dx}lo{g}_{e}f(x)=\frac{f^{\prime }(x)}{f(x)}$

$\int \frac{f^{\prime }(x)}{f(x)}dx=lo{g}_{e}f(x)$