0.2: The Exponential Function and the Natural Logarithm
The transcendental number \(e\), approximately \(2.71828\), is defined as
\[e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n.\nonumber \]
The exponential function \(\exp (x) = e^x\) and natural logarithm \(\ln x\) are inverse functions satisfying
\[e^{\ln x}=x,\quad\ln e^x=x.\nonumber \]
The usual rules of exponents apply:
\[e^xe^y=e^{x+y},\quad e^x/e^y=e^{x-y},\quad (e^x)^p=e^{px}.\nonumber \]
The corresponding rules for the logarithmic function are
\[\ln (xy)=\ln x+\ln y,\quad ln (x/y)=\ln x-\ln y,\quad \ln x^p=p\ln x.\nonumber \]