Answer
$$x = 2$$
Work Step by Step
$$\eqalign{
& {\log _{10}}{x^2} + 3{\log _{10}}x = {\log _{10}}32 \cr
& {\text{Write 32 as }}{{\text{2}}^5} \cr
& {\log _{10}}{x^2} + 3{\log _{10}}x = {\log _{10}}{2^5} \cr
& {\text{Apply the power property }}a\log b = \log {b^a} \cr
& {\log _{10}}{x^2} + {\log _{10}}{x^3} = {\log _{10}}{2^5} \cr
& {\text{Apply the product property }}\log a + \log b = \log ab \cr
& {\log _{10}}\left( {{x^2} \cdot {x^3}} \right) = {\log _{10}}{2^5} \cr
& {\log _{10}}\left( {{x^5}} \right) = {\log _{10}}{2^5} \cr
& {\text{Then }} \cr
& {\text{1}}{{\text{0}}^{{{\log }_{10}}\left( {{x^5}} \right)}} = {\text{1}}{{\text{0}}^{{{\log }_{10}}\left( {{2^5}} \right)}} \cr
& {x^5} = {2^5} \cr
& {\text{Simplifying}} \cr
& x = 2 \cr} $$
