HSC Extension 1 and 2 Mathematics/Exponential and logarithmic functions

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Index laws[edit]

Failed to parse (lexing error): a^x × a^y = a^[(x+y)]

Failed to parse (lexing error): a^x ÷ a^y = a^[(x-y)]


Zero and negative indices[edit]

Rational indices[edit]

Exponential and logarithmic functions[edit]

Logarithms[edit]

Formula for converting one base to another[edit]

Derivative of exponential functions[edit]

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

Evaluation of e[edit]

Natural logarithms[edit]

Derivative of ekx, k a constant[edit]

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

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

Derivative of logex[edit]

\frac{d}{dx}log_ex = \frac{1}{x}, x > 0

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

Derivative of loge(ax), a > 0[edit]

\frac{d}{dx}log_e(ax) = \frac{1}{x}

Derivative of loge(ax+b)[edit]

\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)