HSC Extension 1 and 2 Mathematics/Exponential and logarithmic functions

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

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

Derivative of exponential functions
$$\frac{d}{dx}e^x = e^x$$

Derivative of ekx, k a constant
$$\frac{d}{dx}e^{kx} = ke^{kx}$$

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

Derivative of logex
$$\frac{d}{dx}log_ex = \frac{1}{x}, x > 0$$

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

Derivative of loge(ax), a > 0
$$\frac{d}{dx}log_e(ax) = \frac{1}{x}$$

Derivative of loge(ax+b)
$$\frac{d}{dx}log_e(ax + b) = \frac{a}{ax + b}$$

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

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