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\title{Logarithme népérien}

\section*{Définition}

\begin{minipage}{0.5\linewidth}
La fonction \textit{logarithme népérien} est la fonction notée $\ln$, définie sur \intervOO{0}{+\infty}, telle que:
\[
\ln(\e^x) = \e^{\ln x} = x
\]
pour tout $x>0$.

\bigskip

On dit que les fonctions exponentielle et logarithme népérien sont réciproques.
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\section*{Propriétés importantes}

$a>0$, $b>0$, $n\in\Z*$.

\subsection*{Pour transformer des expressions}
\[
\boxed{\ln(1)=0}
\qquad
\boxed{\ln(ab) = \ln(a)+\ln(b)\strut}
\qquad
\boxed{\ln\left(\frac{a}{b}\right)=\ln(a)-\ln(b)}
\qquad
\boxed{\ln\big(a^n\big)=n\ln(a)}
\qquad
\boxed{\ln\left(\dfrac{1}{a}\right)=-\ln(a)}
\]
\subsection*{Pour résoudre des équations et des inéquations}
\[
\boxed{a < b \iff \ln(a) < \ln(b)}
\qquad
\boxed{a = b \iff \ln(a) = \ln(b)}
\qquad
\boxed{a > b \iff \ln(a) > \ln(b)}
\]

\section*{Dérivation}

\[
\boxed{\ln^\prime(x)=\frac{1}{x}}
\qquad
\boxed{\big(\ln(u)\big)^\prime = \frac{u'}{u}}
\]

\section*{Limites}
\[
\boxed{\limite{x}{0}{\ln(x)}=-\infty}
\qquad
\boxed{\limite{x}{+\infty}{\ln(x)}=+\infty}
\qquad
\boxed{n\in\N*,\ \limite{x}{0}{x^n\ln(x)}=0}
\qquad
\boxed{n\in\N*,\ \limite{x}{+\infty}{\frac{\ln(x)}{x^n}}=0}
\]
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