Fonction logarithme népérien – exercices de maths Terminale

Exercice 1Propriétés

1. \(\ln 27 - \ln 3\)
2. \(\ln 20 + \ln 5 - \ln 4\)
3. \(\ln e^{-2}\)
4. \(e^{\ln 11}\)
5. \(\ln \sqrt[3]{e}\)
6. \(\ln\dfrac{1}{e^5}\)

Exercice 2ln 2 et ln 5

1. \(\ln 10\)
2. \(\ln 40\)
3. \(\ln \dfrac{8}{25}\)
4. \(\ln 0{,}4\)
5. \(\ln \sqrt{50}\)

Exercice 3Équations

1. \(\ln x = -1\)
2. \(\ln(3x + 2) = 0\)
3. \(e^{x} = 8\)
4. \(\ln(x + 2) + \ln(x - 2) = \ln 5\)
5. \(2\ln x = \ln(3x + 4)\)

Exercice 4Inéquations

1. \(\ln x > 1\)
2. \(\ln(2x - 5) \leq 0\)
3. \(e^{3x} < 20\)
4. \(0{,}7^n < 0{,}001\)
5. \(1{,}08^n \geq 3\)

Exercice 5Dérivées

1. \(x^2\ln x\)
2. \(\ln(x^2 + 4)\)
3. \(\dfrac{\ln x}{x^2}\)
4. \((\ln x)^3\)
5. \(\ln(5x - 1)\)

Exercice 6Limites

1. \(\lim_{+\infty} \dfrac{\ln x}{x^2}\)
2. \(\lim_{0^+} x^2\ln x\)
3. \(\lim_{+\infty} x - \ln x\)
4. \(\lim_{0^+} \dfrac{\ln x}{x^2}\)
5. \(\lim_{1} \dfrac{\ln x}{x^2 - 1}\)

Exercice 7Étude

\(f(x) = x^2 - 2\ln x\).

1. Limites ?
2. Variations, minimum ?
3. \(\ln x \leq \dfrac{x^2 - 1}{2}\) ?

Exercice 8ln x / x²

\(g(x) = \dfrac{\ln x}{x^2}\).

1. Limites ?
2. Variations, maximum ?
3. \(g(x) = 0\) ; solutions de \(g(x) = 0{,}1\) ?

Exercice 9Suites

1. Capital à \(5\,\%\) : temps de doublement ?
2. \(u_n = 8 \times 0{,}7^n\) : \(u_n < 10^{-4}\) ?
3. \(2^n > 10^{15}\) ?

Exercice 10Problème

\(f(x) = \ln(x^2 + 1) - \dfrac{x}{2}\).

1. Variations ?
2. Signe ?
3. Convexité ?