Calculs guidés : puissances et racines

Exemple

On pose $\ln 2 = 0{,}7$ et $\ln 3 = 1{,}1$. Calculons $\ln 4$, $\ln 8$, $\ln\sqrt[3]{2}$, $\ln\sqrt{3}$ et $\ln\sqrt[3]{3^5}$.

• $\ln 4 = \ln 2^2 = 2\ln 2 = 2 \times 0{,}7 = 1{,}4 \;\Rightarrow\; \ln 4 = 1{,}4$
• $\ln 8 = \ln 2^3 = 3\ln 2 = 3 \times 0{,}7 = 2{,}1 \;\Rightarrow\; \ln 8 = 2{,}1$
• $\ln\sqrt[3]{2} = \ln 2^{\frac{1}{3}} = \dfrac{1}{3}\ln 2 = \dfrac{1}{3}\times 0{,}7 \approx 0{,}233$
• $\ln\sqrt{3} = \ln 3^{\frac{1}{2}} = \dfrac{1}{2}\ln 3 = \dfrac{1}{2}\times 1{,}1 = 0{,}55 \;\Rightarrow\; \ln\sqrt{3} = 0{,}55$
• $\ln\sqrt[3]{3^5} = \ln 3^{\frac{5}{3}} = \dfrac{5}{3}\ln 3 = \dfrac{5}{3}\times 1{,}1 \approx 1{,}833$