Rewrite the argument of the logarithm in exponential form.

The argument is $3\sqrt{3}$.

$$3\sqrt{3} = 3^1 \cdot 3^{1/2}$$

Using the exponent rule $a^m \cdot a^n = a^{m+n}$:

$$3^1 \cdot 3^{1/2} = 3^{1 + \frac{1}{2}} = 3^{\frac{2}{2} + \frac{1}{2}} = 3^{\frac{3}{2}}$$

$$\log_{10} (3^{\frac{3}{2}}) = \frac{3}{2} \log_{10} 3$$

The simplified expression is:

$$\frac{3}{2} \log_{10} 3$$