Answer
$1+\frac{1}{2}\log_3{x}$
Work Step by Step
Note that:
\begin{align*}
\sqrt{9x} &= \sqrt{3^2x}
\\&=3\sqrt{x}
\\&=3\cdot x^{\frac{1}{2}}
\end{align*}
Thus, the given expression is equivalent to:
$$\log_3{\left(3x^{\frac{1}{2}}\right)}$$
Recall:
(1) Product Property of Logarithms: $\log_a{b}+\log_a{c}=\log_a{bc}$.
(2) Power Property of Logarithms: $\log_a{b^n} = n\cdot \log_a{b}$
Use the Product Property to obtain:
\begin{align*}
\log_3{\left(3x^{\frac{1}{2}}\right)}&=\log_3{3} + \log_3{x^{\frac{1}{2}}}\\
\end{align*}
Use the Power Property to obtain:
\begin{align*}
\log_3{3} + \log_3{x^{\frac{1}{2}}}&=\log_3{3} +\frac{1}{2}\log_3{x}\\
\end{align*}
Note that $\log_3{3}=1$.
Thus,
\begin{align*}
\log_3{3} +\frac{1}{2}\log_3{x}&=1+\frac{1}{2}\log_3{x}\\
\end{align*}
