HSC Extension 1 and 2 Mathematics/Exponential and logarithmic functions

Index laws

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

$a^{x}\div a^{y}=a^{(x-y)}$

Logarithms

Derivative of exponential functions

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

Derivative of e^kx, k a constant

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

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

Derivative of log_ex

${\frac {d}{dx}}log_{e}x={\frac {1}{x}},x>0$

$\int {\frac {1}{x}}dx=log_{e}x+c$

Derivative of log_e(ax), a > 0

${\frac {d}{dx}}log_{e}(ax)={\frac {1}{x}}$

Derivative of log_e(ax+b)

${\frac {d}{dx}}log_{e}(ax+b)={\frac {a}{ax+b}}$

${\frac {d}{dx}}log_{e}f(x)={\frac {f'(x)}{f(x)}}$

$\int {\frac {f'(x)}{f(x)}}dx=log_{e}f(x)$