The natural logarithm.

Definition A..1.1   The natural logarithm function is the function denoted $ \ln$ defined by the following conditions:
  1. $ \ln$ is differentiable on $ (0,+\infty)$ and $ \forall x \in (0,+\infty),
(\ln )'(x)= \frac 1x$ .
  2. $ \ln 1 =0$ .

Theorem A..1.2   $ \forall x,y \in (0,+\infty), \; \ln (xy) = \ln x+ \ln y$ .

Corollary A..1.3   $ \forall x,y \in (0,+\infty), \; \ln (\frac xy) = \ln x - \ln y.$

Example A..1.4   $ \ln 23 = \ln 2- \ln3$ .

Corollary A..1.5   $ \forall x,\in (0,+\infty), \; \forall \alpha \in \mathbb{Q}, \; \ln (x^{\alpha}) = \alpha \ln x$ .

Example A..1.6   $ \ln \sqrt{48} = \ln (2^4 \cdot 3)^{1/2} = \ln 2^2 \cdot 3^(1/2) = 2 \ln 2 + \frac 12 \ln 3$ .

Figure 1: The graph of the natural logarithm.
\begin{figure}\mbox{\epsfig{file=ln.eps,height=6cm}}\end{figure}



Noah Dana-Picard 2007-12-28