[tex]\displaystyle\bf\\3^\frac{x+1}{2}} =2\cdot2^x\\\\3^\frac{x+1}{2}} =2^{x+1}\\\\3^{\frac{1}{2}\cdot(x+1)}=2^{x+1}\\\\\Big(\sqrt{3}\Big)^{x+1}=2^{x+1}~~~\Big|~~:2^{x+1}\\\\\frac{\Big(\sqrt{3}\Big)^{x+1}}{2^{x+1}}=1\\\\\\\left(\frac{\sqrt{3}}{2}\right)^{x+1}=1\\\\\\\textbf{Orice numar rigicat la puterea }0=1\\\\\implies~~x+1=0\\\\\implies~~\boxed{\bf~x=-1}[/tex]