Exercice 2 Simplifier les expressions suivantes : A=ln(3)+ln(13)−ln(9)A = \ln(\sqrt{3}) + \ln\left(\dfrac{1}{3}\right) - \ln(9)A=ln(3)+ln(31)−ln(9) B=ln(2+1)+ln(2−1)B = \ln(\sqrt{2}+1) + \ln(\sqrt{2}-1)B=ln(2+1)+ln(2−1) Correction A=ln(3)+ln(13)−ln(9)=ln(312)−ln(3)−ln(32)=12ln(3)−ln(3)−2ln(3)=(12−1−2)ln(3)=−52ln(3)\begin{align*} A &= \ln\left(\sqrt{3}\right)+\ln\left(\dfrac{1}{3}\right)-\ln(9) \\ &= \ln\left(3^{\frac{1}{2}}\right) - \ln(3) - \ln(3^2) \\ &= \frac{1}{2} \ln(3) - \ln(3) - 2 \ln(3) \\ &= \left(\frac{1}{2} - 1 - 2\right) \ln(3) = \frac{-5}{2} \ln(3) \end{align*}A=ln(3)+ln(31)−ln(9)=ln(321)−ln(3)−ln(32)=21ln(3)−ln(3)−2ln(3)=(21−1−2)ln(3)=2−5ln(3) B=ln(2+1)+ln(2−1)=ln[(2+1)(2−1)]=ln(22−12)=ln(1)=0\begin{align*} B &= \ln(\sqrt{2}+1) + \ln(\sqrt{2}-1) \\ &= \ln\left[(\sqrt{2}+1)(\sqrt{2}-1)\right] \\ &= \ln(\sqrt{2}^2 - 1^2) = \ln(1) = 0 \end{align*}B=ln(2+1)+ln(2−1)=ln[(2+1)(2−1)]=ln(22−12)=ln(1)=0