Exercice 1
Simplifier \(A = \ln8 - 2\ln2 + \ln\frac12\) et \(B = \ln(e^3) - \ln\sqrt{e}\).
Correction
\(A = 3\ln2 - 2\ln2 - \ln2 = 0\). \(B = 3 - \frac12 = \frac52\).
Exercice 2
Résoudre : \(\ln(2x - 3) = 0\) ; \(\ln(x^2 - 4) = \ln(3x)\) ; \(\ln(x - 2) < 1\).
Correction
\(2x - 3 = 1\) : \(x = 2\). Domaine \(x > 2\) : \(x^2 - 3x - 4 = 0\), \(x = 4\). Domaine \(x > 2\) : \(x - 2 < e\), \(S = ]2 ; 2 + e[\).
Exercice 3
Calculer : \(\lim_{x\to+\infty}\frac{\ln x}{x^2}\), \(\lim_{x\to0^+}(x - \ln x)\), \(\lim_{x\to+\infty}\ln\frac{x + 1}{x}\).
Correction
\(\frac{\ln x}{x}\cdot\frac1x\to0\). \(+\infty\). \(\ln1 = 0\).
Exercice 4
Dériver \(f(x) = x\ln x - x\) et \(g(x) = \ln(x^2 + x + 1)\).
Correction
\(f'(x) = \ln x + 1 - 1 = \ln x\). \(g'(x) = \frac{2x + 1}{x^2 + x + 1}\).
Exercice 5
Résoudre \((\ln x)^2 + \ln x - 2 = 0\).
Correction
\(\ln x = 1\) ou \(\ln x = -2\) : \(x = e\) ou \(x = e^{-2}\).