1.2: The Exponential Function and the Natural Logarithm
The transcendental number \(e\), approximately \(2.71828\), is defined as \[e=\underset{n\to\infty}{\lim}\left(1+\frac{1}{n}\right)^{n}.\nonumber\]
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 \[\ln(xy)=\ln x+\ln y,\quad \ln (x/y)=\ln x-\ln y,\quad \ln x^p=p\ln x.\nonumber\]