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Section 1.11 The natural logarithm
Definition 1.11.1 . Natural logarithm.
The natural logarithm function is the function \(\ln\) with domain \(D=(0,\infty)\) defined as
\begin{equation*}
\ln x=\int_1^x\frac{1}{t}\, dt\text{.}
\end{equation*}
Theorem 1.11.2 . Natural logarithm properties.
The natural logarithm is differentiable (hence also continuous) on \((0,\infty)\) and satisfies
\begin{equation*}
\frac{d}{dx}\ln x=\frac{1}{x}\text{.}
\end{equation*}
for all \(x\) in \((0,\infty)\text{.}\)
The natural logarithm is increasing on \((0,\infty)\) and hence one-to-one. The graph of \(\ln\) is always concave down.
We have
\begin{align}
\lim_{x\to\infty}\ln x\amp =\infty\tag{1.11.1}\\
\lim_{x\to 0^+}\ln x\amp =-\infty\tag{1.11.2}
\end{align}
The range of \(\ln\) is \((-\infty, \infty)\text{.}\)
\(\ln 1=0\text{.}\)
We have
\begin{align}
\ln(ab)\amp =\ln a+\ln b, \text{ for all \(a,b\in (0,\infty)\). }\tag{1.11.3}\\
\ln(a/b)\amp =\ln a-\ln b, \text{ for all \(a,b\in (0,\infty)\), \(b\ne 0\). }\tag{1.11.4}\\
\ln a^r\amp =r\ln a, \text{ for all \(a\in (0,\infty)\) and \(r\in \R\). }\tag{1.11.5}
\end{align}
Corollary 1.11.4 . Antiderivative of \(f(x)=1/x\) .
The function \(F(x)=\ln\lvert x\rvert\) is an antiderivative of the function \(f(x)=1/x\) on \(D=(-\infty, 0)\cup (0,\infty)\text{:}\) i.e., we have
\begin{equation*}
\int\frac{1}{x}\, dx=\ln \vert x\rvert\, dx+C\text{.}
\end{equation*}
Definition 1.11.5 . Euler’s number.
Euler’s number , denoted \(e\text{,}\) is the unique number in \(D=(0,\infty)\) satisfying \(\ln e=1\text{.}\) In other words, \(e\) is the number satisfying
\begin{equation*}
1=\int_1^e\frac{1}{t}\, dt\text{.}
\end{equation*}
Theorem 1.11.6 . Trigonometric antiderivatives.
\(\displaystyle \displaystyle\int \tan x\, dx=-\ln\lvert \cos x\rvert+C=\ln\lvert\sec x\rvert+C\)
\(\displaystyle \displaystyle\int \cot x\, dx=\ln\vert \sin x\vert+C\)
\(\displaystyle \displaystyle\int \sec x\, dx=\ln\vert \sec x+\tan x\vert+C\)
\(\displaystyle \displaystyle\int \csc x\, dx=-\ln\vert \csc x+\cot x\vert+C\)
