Answer
$\log_\sqrt{3}{9} = 4$
Work Step by Step
Let
$\log_\sqrt{3}{9}=y$
RECALL:
$\log_a{x} = y \longrightarrow a^y=x$
Use the rule above to obtain:
$\log_\sqrt3{9} = y \longrightarrow (\sqrt{3})^y=9$
Write $9$ as $3^2$ to obtain:
$(\sqrt{3})^y=3^2$
Note that $\sqrt{3} = 3^{\frac{1}{2}}$. Thus, the equation above is equivalent to:
$(3^{\frac{1}{2}})^y = 3^2$
Use the rule $(a^m)^n=a^{mn}$ to obtain:
$3^{\frac{1}{2} \cdot y} = 3^2
\\3^{\frac{y}{2}} = 3^2$
$\frac{y}{2} = 2$
Multiply $2$ to both sides of the equation to obtain:
$2(\frac{y}{2}) = 2(2)
\\y = 4$
Thus,
$\log_\sqrt{3}{9} = 4$
