[tex]\Large\boxed{\begin{array}{l}\sf \log_4\sqrt{32}=x\implies 4^x=\sqrt{32}\\\sf (2^2)^x=\sqrt{2^5}\\\sf 2^{2x}=2^{\frac{5}{2}}\\\sf 2x=\dfrac{5}{2}\\\\\sf 4x=5\\\sf x=\dfrac{5}{4}\end{array}}[/tex]
