6.4 The Natural Logarithm
In the last section we studied the natural exponential function.
\[
\begin{align}
p &\mathrel{\overset{\exp}{\leftarrow}} e
\\
p &= \exp e
\end{align}
\]
The natural exponential takes exponents, \( e \), as input and produces powers, \( p \), as output. The natural logarithm inverts the relationship between \( p \) and \( e \).
\[
\begin{align}
e &\mathrel{\overset{\log}{\leftarrow}} p
\\
e &= \log p
\end{align}
\]
Since the natural logarithm is an inverse, it cancels with the natural exponential.
\[
\begin{align}
\log \exp e &= e
&
&\andSpaced
&
\exp \log p &= p
\end{align}
\]
We need one restriction: the natural exponential function produces positive output, \( p \gt 0 \), only, so the natural logarithm will only accept positive input.
Theorem.
The natural logarithm satisfies the laws
\[
\begin{align}
&\log 1 = 0
\\
&\log \naturalbase = 1
\\
&\log (p_1 \mult p_2) = \log p_1 + \log p_2
\end{align}
\]
where \( p_1 : \Number \) and \( p_2 : \Number \) are positive numbers.
Why?
The first two laws are easy enough cancellations.
\[
\begin{align}
&\log 1 = \log \exp 0 = 0
\\
&\log \naturalbase = \log \exp 1 = 1
\end{align}
\]
For the third we define
\[
\begin{align}
e_1 &= \log p_1
&
&\andSpaced
&
e_2 &= \log p_2
\end{align}
\]
so that we can rewrite the analogous exponential law.
\[
\begin{align}
&\exp (e_1 + e_2) = \exp e_1 \mult \exp e_2
\\
&\exp (\log p_1 + \log p_2) = \exp \log p_1 \mult \exp \log p_2
\\
&\exp (\log p_1 + \log p_2) = p_1 \mult p_2
\\
&\log \exp (\log p_1 + \log p_2) = \log (p_1 \mult p_2)
\\
&\log p_1 + \log p_2 = \log (p_1 \mult p_2)
\end{align}
\]
We've found it!
We can graph the natural logarithmic function by plotting points. Each point \( (e, p) \) on the natural exponential's graph, is a point \( (p, e) \) on the natural logarithm's graph.
| \( p \) | \( e \) |
|---|---|
| \( \naturalbase^{-2} \) | \( -2 \) |
| \( \naturalbase^{-1} \) | \( -1 \) |
| \( \naturalbase^{0} \) | \( 0 \) |
| \( \naturalbase^{1} \) | \( 1 \) |
| \( \naturalbase^2 \) | \( 2 \) |
See if you can draw transition diagrams for the graph's height, slope, and bend!