From 90cadf87ce05636e34a69c248a9119a63c4e4cda Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Mon, 4 Jan 2021 15:03:01 +0100 Subject: [PATCH] =?UTF-8?q?Feat:=202E=20sur=20les=20relations=20alg=C3=A9b?= =?UTF-8?q?riques=20du=20log?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../1E_equations.tex | 6 +- .../2B_rel_fonctionnelles.pdf | Bin 0 -> 60302 bytes .../2B_rel_fonctionnelles.tex | 39 +++++++++ .../2E_puiss_10.pdf | Bin 0 -> 80946 bytes .../2E_puiss_10.tex | 21 +++++ .../exercises.tex | 76 ++++++++++++++++++ .../index.rst | 16 +++- 7 files changed, 154 insertions(+), 4 deletions(-) create mode 100644 TST/07_Logarithme_et_equation_puissance/2B_rel_fonctionnelles.pdf create mode 100644 TST/07_Logarithme_et_equation_puissance/2B_rel_fonctionnelles.tex create mode 100644 TST/07_Logarithme_et_equation_puissance/2E_puiss_10.pdf create mode 100644 TST/07_Logarithme_et_equation_puissance/2E_puiss_10.tex diff --git a/TST/07_Logarithme_et_equation_puissance/1E_equations.tex b/TST/07_Logarithme_et_equation_puissance/1E_equations.tex index 1577b93..e79a806 100644 --- a/TST/07_Logarithme_et_equation_puissance/1E_equations.tex +++ b/TST/07_Logarithme_et_equation_puissance/1E_equations.tex @@ -2,8 +2,8 @@ \usepackage{myXsim} \author{Benjamin Bertrand} -\title{Logarithme et equation puissance - Cours} -\date{décembre 2020} +\title{Logarithme et équation puissance - Cours} +\date{Janvier 2021} \DeclareExerciseCollection{banque} \xsimsetup{ @@ -15,4 +15,4 @@ \input{exercises.tex} \printcollection{banque} -\end{document} \ No newline at end of file +\end{document} diff --git a/TST/07_Logarithme_et_equation_puissance/2B_rel_fonctionnelles.pdf b/TST/07_Logarithme_et_equation_puissance/2B_rel_fonctionnelles.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3a33bbc1f5fb58563d0582d7b19f6cc6a45c1d70 GIT binary patch literal 60302 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+\setcounter{section}{1} +\section{Fonction logarithme (suite)} + +\begin{propriete}{Relations fonctionnelles} +Soient $a$ et $b$ deux nombres réels strictement positifs et $n$ un entier naturel. +\begin{align*} + \ln(a \times b) &= \ln(a) + \ln(b)\\ + \ln(a^n) &= n\ln(a) \\ + \ln\left( \frac{a}{b} \right) &= \ln(a) - \ln(b) \\ + \ln\left( \frac{1}{a} \right) &= - \ln(a) \\ +\end{align*} +\end{propriete} + +\paragraph{Exemple} + +\subsection*{Exemples} +Écrire avec un seul logarithme le nombre +\[ +A = 3\ln(8) - \ln(2) + 4\ln(5) +\] +\afaire{Utiliser les formules de la propriété pour simplifier le calcul} + + + + +\end{document} diff --git a/TST/07_Logarithme_et_equation_puissance/2E_puiss_10.pdf b/TST/07_Logarithme_et_equation_puissance/2E_puiss_10.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a666325be7220f1c16da54082140f6098c6d0204 GIT binary patch literal 80946 zcmce5V~{A_vSr)0ZQHhO+qUiQ)AniGwr$%yZQI>*zWd%xOvL*!^XJ}*sH)73+_@rm 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+\author{Benjamin Bertrand} +\title{Logarithme et équation puissance - Cours} +\date{Janvier 2021} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=2, +} +\setlength{\columnseprule}{0pt} + +\begin{document} + +\input{exercises.tex} +\printcollection{banque} +\vfill +\printcollection{banque} + +\end{document} diff --git a/TST/07_Logarithme_et_equation_puissance/exercises.tex b/TST/07_Logarithme_et_equation_puissance/exercises.tex index 81c89c5..eb942cf 100644 --- a/TST/07_Logarithme_et_equation_puissance/exercises.tex +++ b/TST/07_Logarithme_et_equation_puissance/exercises.tex @@ -70,4 +70,80 @@ \end{enumerate} \end{exercise} +\begin{exercise}[subtitle={Résolution d'équations}, step={2}, origin={Création}, topics={Logarithme et equation puissance}, tags={logarithme, fonctions}] + Résoudre les équations suivantes + \begin{multicols}{4} + \begin{enumerate} + \item $10^{x} = 200$ + \item $10^{x} = 2$ + + \item $10^{x} = -10$ + \item $10^{2x} = 3$ + + \item $10^{-3x} = 10$ + \item $10^{5x+1} = 10$ + + \item $2\times10^{x} = 6$ + \item $-3\times10^{x} = -9$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Résolution d'inéquations}, step={2}, origin={Création}, topics={Logarithme et equation puissance}, tags={logarithme, fonctions}] + Résoudre les inéquations suivantes + \begin{multicols}{4} + \begin{enumerate} + \item $10^{x} \leq 300$ + \item $10^{x} > 45$ + + \item $10^{x} < 100$ + \item $10^{3x} \geq 3$ + + \item $10^{-0.1x} \leq 10$ + \item $10^{2x+1} \geq 5$ + + \item $3\times10^{x} > 6$ + \item $-2\times10^{x} < -8$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Relation fonctionnelle}, step={2}, origin={Création}, topics={Logarithme et equation puissance}, tags={logarithme, fonctions}] + \begin{enumerate} + \item Calculer les quantités suivantes arrondis au millième. + \begin{multicols}{3} + \begin{enumerate} + \item $A = \ln(6)$ + \item $B = \ln(32)$ + \item $C = \ln(21)$ + \item $D = \ln(27)$ + + \item $E = \ln(2) + \ln(3)$ + \item $F = \ln(3) + \ln(7)$ + \item $G = \ln(2) + \ln(16)$ + \item $H = \ln(63) - \ln(3)$ + + \item $I = \ln(108) - \ln(4)$ + \item $J = 5\ln(2)$ + \item $K = 3\ln(3)$ + \item $L = - \ln(\frac{1}{6})$ + \end{enumerate} + \end{multicols} + \item Conjecture des formules ci-dessous + \[ + \log(a) + \log(b) = \log(...) \qquad \qquad + \log(a) - \log(b) = \log(...) \qquad \qquad + n\log(a) = \log(...) + \] + + \begin{multicols}{2} + \item (*) Soient $x$ et $y$ strictement positif. Après avoir calculer séparément $e^{\ln(x) + \ln(y)}$ et $e^{\ln(x\times y)}$, démontrer que $\ln(x \times y) = \ln(x) + \ln(y)$. + \item (*) Démontrer que pour tout $n \in \N$, $\ln(a^n) = n \ln(a)$. + \item (*) Démontrer que $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$. + \item (*) En déduire une formule pour $\ln(\frac{1}{a})$ + + \end{multicols} + \end{enumerate} +\end{exercise} + \collectexercisesstop{banque} diff --git a/TST/07_Logarithme_et_equation_puissance/index.rst b/TST/07_Logarithme_et_equation_puissance/index.rst index 71adab0..6c18347 100644 --- a/TST/07_Logarithme_et_equation_puissance/index.rst +++ b/TST/07_Logarithme_et_equation_puissance/index.rst @@ -2,7 +2,7 @@ Logarithme et équation puissance ################################ :date: 2020-12-17 -:modified: 2021-01-02 +:modified: 2021-01-04 :authors: Benjamin Bertrand :tags: Logarithme, Fonctions :category: TST @@ -31,6 +31,20 @@ Bilan: On pose l'inéquation du seuil et on introduit la fonction log décimal Étape 2: Premières équations/inéquations ======================================== +Les élèves commencent à utiliser le logarithme pour résoudre des équations/inéquations avec les puissances de 10. Les exercices 1 et 2 sont à faire en même temps et en colonne pour que la difficulté soit progressive. + +On motivera l'exercice 3 par la volonté d'apprendre à maitriser le log et résoudre des équations puissances de bases différentes de 10. + +.. image:: ./2E_puiss_10.pdf + :height: 200px + :alt: Exercices d'équations et inéquations avec des puissances de 10. + +Bilan de l'exercice 3 à recopier dans le cours sur les formules algébriques du log + +.. image:: ./2B_rel_fonctionnelles.pdf + :height: 200px + :alt: Cours sur les relations algébriques du logarithme. + Étape 3: Manipulation algébrique du log =======================================