Determine whether the value x=729 is a solution of the equation ${\log }_9(7x)=\frac{3}{2}$ .

A) not a solution
B) solution

To determine if $x=729$ is a solution, substitute $x$ into the equation: ${\log }_9(7(729))=\frac{3}{2}$ . Simplify the inside of the logarithm: $7(729)=5103$ . The equation becomes ${\log }_9(5103)=\frac{3}{2}$ . The right side, $\frac{3}{2}$ , suggests that $9^{\frac{3}{2}}$ should equal the argument of the logarithm for the equation to hold true. However, $9^{\frac{3}{2}}=(9^1{)}^{\frac{3}{2}}=9^{1\cdot \frac{3}{2}}=9^{\frac{3}{2}}=27$ , which is not equal to 5103. Therefore, $x=729$ is not a solution to the given equation.