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\begin{document}
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\begin{center}
    \begin{tabularx}{\textwidth}{@{} CCC @{}}
        %\toprule
            \multirow{2}{*}{\parbox{\linewidth}{Prof MOSAID \newline \mylink }}
            & Serie - Fonctions logarithmiques & \hfill 2BAC-PC/SVT \\
        \bottomrule
    \end{tabularx}
\end{center}
\textbf{\underline{Exercice 1:}}\\
\noindent
\begin{tabular}{@{}p{0.01\textwidth}|p{0.33\textwidth}p{0.33\textwidth}p{0.33\textwidth}}
        &\multicolumn{3}{l}{Pour chaque fonction suivante, déterminez le domaine de définition:} \\
    &\begin{enumerate}[topsep=6pt, partopsep=0pt, parsep=0pt, itemsep=0pt,
            after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( f(x) = \ln(x) \)
        \item \( g(x) = \ln(2x + 1) \)
        \item \( h(x) = \ln(x^2 - 4) \)
    \end{enumerate}
    &\begin{enumerate}[topsep=6pt, partopsep=0pt, parsep=0pt, itemsep=0pt,start=4,
            after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( m(x) = \ln(3x^2 + 5x + 2) \)
        \item \( n(x) = \ln(\sqrt{x}) \)
        \item \( k(x) = \ln\left(\frac{1}{x - 3}\right) \)
    \end{enumerate}
    &\begin{enumerate}[topsep=6pt, partopsep=0pt, parsep=0pt, itemsep=0pt,start=7,
            after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( q(x) = \ln(|x|) \)
        \item \( r(x) = \ln(x^3 - 8x) \)
        \item \( s(x) = \ln(4x^2 - 16) \)
    \end{enumerate}\\
\end{tabular}
\\
\\
\textbf{\underline{Exercice 2:}}\\
\noindent
\begin{tabular}{@{}p{0.01\textwidth}|p{0.25\textwidth}p{0.25\textwidth}p{0.35\textwidth}}
        &\multicolumn{3}{l}{Simplifer et calculer les expressions suivantes:} \\
    &\begin{itemize}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=5pt,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( \ln\left(\frac{2 \times 10^3}{\sqrt{5}}\right) \)
        \item \( \ln\left(\frac{10^2 \times 10}{\sqrt{10^3}}\right) \)
    \end{itemize}
    &\begin{itemize}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=5pt,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( \ln\left(\frac{10^4 \times 3^{2\ln(7)}}{5 \times 10^{\ln(2)}}\right) \)
        \item \( \ln\left(\frac{\ln(8) + \ln(10)}{\ln(2) + \ln(5)}\right) \)
    \end{itemize}
    &\begin{itemize}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=5pt,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( \ln\left(\frac{3^{\ln(4)} \times 9^{\ln(9)}}{2^{\ln(2)}\times 27^{\ln(3)}}\right) \)
        \item \( \ln\left(\frac{3^{2\ln(3)} \times 5^{\ln(5)} \times 7^{\ln(7)}}{2^{\ln(2)} \times 11^{\ln(11)} \times 13^{\ln(13)}}\right) \)
    \end{itemize}\\
\end{tabular}
\\
\\
\textbf{\underline{Exercice 3:}}\\
\noindent
\begin{tabular}{@{}p{0.01\textwidth}|p{0.30\textwidth}p{0.30\textwidth}p{0.30\textwidth}}
        &\multicolumn{3}{l}{Résoudre les équations et inéquations suivantes:} \\
    &\begin{enumerate}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=0pt,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( \ln(2x + 1) = 5 \)
        \item \( \ln(x^2 - 4) = 0 \)
        \item \( \ln\left(\frac{1}{x - 3}\right) = -2 \)
    \end{enumerate}
    &\begin{enumerate}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=0pt, start=4,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( \ln(3x^2 + 5x + 2) = 4 \)
        \item \( 2\ln(x) = 6 \)
        \item \( \ln(|x|) = -1 \)
    \end{enumerate}
    &\begin{enumerate}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=0pt, start=7,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( \ln(x^2 - 8x) < \ln(4) \)
        \item \( \ln^2(x)+3\ln(x)-4 \le 0 \)
        \item \( \ln(x - 2)- \ln(x-4) < 0 \)

    \end{enumerate}\\
\end{tabular}
\\
\\
\textbf{\underline{Exercice 4:}}\\
\noindent
\begin{tabular}{@{}p{0.01\textwidth}|p{0.30\textwidth}p{0.30\textwidth}p{0.30\textwidth}}
        &\multicolumn{3}{l}{Calculer les limites suivantes:} \\
    &\begin{enumerate}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=0pt,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( \displaystyle \lim_{x \to 0} \frac{\ln(1 + x)}{x} \)
        \item \( \displaystyle \lim_{x \to 1^-} \frac{\ln(x)}{x - 1} \)
    \end{enumerate}
    &\begin{enumerate}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=0pt, start=4,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( \displaystyle \lim_{x \to +\infty} x(\ln(x + 1) - \ln(x)) \)
        \item \( \displaystyle \lim_{x \to 1} \frac{\ln(x^2 + 1)}{x - 1} \)
    \end{enumerate}
    &\begin{enumerate}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=0pt, start=7,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( \displaystyle \lim_{x \to 0^+} x\ln(x^2 + 1) \)
        \item \( \displaystyle \lim_{x \to +\infty} \frac{\ln(x^2 + 1)}{\sqrt{x}} \)
    \end{enumerate}\\
\end{tabular}
\\
\\
\textbf{\underline{Exercice 5:}}\\
\noindent
\begin{tabular}{@{}p{0.01\textwidth}|p{0.33\textwidth}p{0.33\textwidth}p{0.33\textwidth}}
        &\multicolumn{3}{l}{Calculer les fonctions dérivées:} \\
    &\begin{enumerate}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=0pt,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
            \item \( f(x) = \ln(2x + 1) \)
            \item \( g(x) = \ln\left(\frac{1}{x - 3}\right) \)
    \end{enumerate}
    &\begin{enumerate}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=0pt, start=3,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
            \item \( h(x) = \ln(3x^2 + 5x + 2) \)
            \item \( k(x) = \ln(\sqrt{x}) \)
    \end{enumerate}
    &\begin{enumerate}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=0pt, start=5,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
            \item \( l(x) = \ln(|x|) \)
            \item \( m(x) = \ln(x^3 - 8x) \)
    \end{enumerate}\\
\end{tabular}
\\
\\
\textbf{\underline{Exercice 6:}}\\
\noindent
\begin{tabular}{@{}p{0.01\textwidth}|p{0.25\textwidth}p{0.25\textwidth}p{0.35\textwidth}}
        &\multicolumn{3}{l}{Etudier et représenter les fonctions suivantes:} \\
    &\begin{itemize}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=5pt,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( f(x) =\ln(\frac{x+3}{2x-1}) \)
        \item \( g(x) = \ln(\ln(x)) \)
    \end{itemize}
    &\begin{itemize}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=5pt,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( h(x) = x^2-\ln(x+1) \)
        \item \( k(x) = \ln\left(\frac{3x}{x^2+1}\right) \)
    \end{itemize}
    &\begin{itemize}[topsep=3pt, partopsep=0pt, parsep=0pt, itemsep=5pt,
        after=\vspace*{-\baselineskip}, before=\vspace*{-\baselineskip}]
        \item \( m(x) = \ln(4x^3-5x+2) \)
            \item \( n(x) = \ln(\sqrt{x+5}) \)
    \end{itemize}\\
\end{tabular}
\\
\\
\textbf{\underline{Exercice 7:}}\\
\noindent
\begin{minipage}[t]{0.48\linewidth}
    I. Soit la fonction \begin{small}\(g\)\end{small} définie sur
        \begin{small}\([0,+\infty[\)\end{small} par:
            \hspace*{1cm}   \begin{small}\(g(x) = 2x^2+1-\ln(x)\)\end{small} \\
    1. Calculer \begin{small}$\displaystyle \lim_{x \to 0^+}g(x)$\end{small} et montrer que
    \begin{small} $\displaystyle \lim_{x \to +\infty} g(x)=+\infty$ \end{small}\\
    2.1 Calculer \(g'(x)\) pour tout \(x \in ]0,+\infty[\)\\
    2.2 Etablir le tableau des variations de \(g\)\\
    2.3 Montrer que \(\forall x \in ]0,+\infty[ \quad g(x)>0\)\\
    \\
    II. Soit la fonction définie sur \(]0,+\infty[\) par \\
    \hspace*{1cm}\(f(x)=2x-2+\frac{\ln(x)}{x}\)\\
    1. Calculer
    $\displaystyle \lim_{x \to 0^+} f(x) \quad$ et
    $\displaystyle \lim_{x \to +\infty} f(x)$\\
    2. Determiner les asymptotes de la courbe \(\mathscr{C_f}\)\\
    3. Monter que \(\forall x \in ]0,+\infty[ \quad f'(x)=\frac{1}{x^2}g(x)\)\\
    4. Etablir le tableau des variations de la fonction \(f\)\\
    5. Calculer \(f"(x)\) et determiner les points d'inflexion\\
    \hspace*{1cm}de  la courbe \(\mathscr{C_f}\)\\
    6. Construir la courbe \(\mathscr{C_f}\) dans un repère orthonormé \\
    \hspace*{0.30cm}\((O,\overrightarrow{i},\overrightarrow{j})\).
    \hspace*{0.20cm} Données: \(\quad e^\frac{3}{2}=4.5 \quad et \quad f(e^\frac{3}{2})=7.3\)
\end{minipage}
\hspace*{0.1cm}
\vline
\hspace*{0.1cm}
\begin{minipage}[t]{0.48\linewidth}
    \vspace*{-0.65cm}
    \textbf{\underline{Exercice 8:}}\\
    I. Soit la fonction \begin{small}\(g\)\end{small} définie sur
        \begin{small}\([0,+\infty[\)\end{small} par:
            \hspace*{1cm}   \begin{small}\(g(x) = x^3-x+1-2\ln(x)\)\end{small} \\
    1. Montrer que \(g'(x)=\frac{(x-1)(x^2+3x+2)}{x}\)\\
    2.1 Calculer \begin{small}$\displaystyle \lim_{x \to 0^+}g(x)$\end{small} et montrer que
    \begin{small} $\displaystyle \lim_{x \to +\infty} g(x)=+\infty$ \end{small}\\
    2.2 Etablir le tableau des variations de \(g\)\\
    2.3 En déduir le signe de \(g(x)\) sur l'intervalle \(]0,+\infty[\)\\
    \\
    II. Soit la fonction définie sur \(]0,+\infty[\) par \\
    \hspace*{1cm}\(f(x)=2x-2+\frac{\ln(x)}{x}\)\\
    1. Calculer \begin{small}$\displaystyle \lim_{x \to 0^+}f(x)$\end{small} et
    \begin{small} $\displaystyle \lim_{x \to +\infty} f(x)$ \end{small}\\
    2. Determiner les deux branches infinies de  \(\mathscr{C_f}\) \\
    3.1 Montrer que \(\forall x \in ]0,+\infty[ \quad f'(x)=\frac{g(x)}{x^3}\)\\
    3.2 Etablir le tableau des variations de \(f\)\\
    4. Determiner l'équation de la tangente \((T)\) à  \(\mathscr{C_f}\) au point \(x_0=1\)\\
    5. Construir  \(\mathscr{C_f}\) dans un repère orthonormé
        \((O,\overrightarrow{i},\overrightarrow{j})\)\\
    6. Déterminer une fonction primitive de la fonction\\
    \Centering \(h(x)=x^2+\frac{1}{x+1}\)
\end{minipage}
\\
\textbf{\underline{Exercice 9:}}\\
\noindent
\begin{tabular}{@{}p{0.01\textwidth}|p{0.98\textwidth}}
    & Simplifier le nombre suivant:
    \(A=\log(\frac{4}{x})-\log(\frac{x}{5})-2\log(\frac{\sqrt{x}}{2})+\log(\frac{25}{x})+\log(50x)\)\\
\end{tabular}
\\
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