Exponential and logarithmic functions

The natural logarithm

The natural logarithm is defined as having its derivative equal to \(\frac{1}{x}\). For \(x > 0\), then

\[ \frac{d}{dx} \ln x = \frac{1}{x}. \]

\[ \lim_{h \to 0} \frac{\ln (1+h)}{h} = 1 \]

The exponential function is defined as the inverse of the natural logarithm

\[ \ln e^x = x. \]

Furthermore \(e\) may be defined by,

\[ \begin{array}{ll} \lim_{n \to \infty} (1 + \frac{1}{n})^n = e, \\ \lim_{n \to \infty} (1 + \frac{x}{n})^n = e^x. \end{array} \]

The derivative of \(y = e^x\) may be calculated by implicit differentation:

\[ \begin{array}{ll} y = e^x &\implies x = \ln y, \\ &\implies 1 = \frac{1}{y} \frac{dy}{dx}, \\ &\implies \frac{dy}{dx} = y = e^x. \end{array} \]

\[ \lim_{h \to 0} \frac{e^h - 1}{h} = 1 \]