For $x, a>0$ the root(s) of the equation ${\log}{ax}a+{\log}{ax}{a}^{2}+{\log}_{{a}^{2}x}{a}^{3}=0$ is (are) given by

Using the change of base formula, rewrite the logarithms as $1/(1 + \log_a x) + 2/(1 + \log_a x) + 3/(2 + \log_a x) = 0$. Let $y = \log_a x$, turning the equation into $3/(1 + y) + 3/(2 + y) = 0$, which simplifies to $3(3 + 2y) = 0$ and yields $y = -3/2$. Since $y = \log_a x = -3/2$, converting this back to exponential form gives the root as $x = a^{-3/2}$.

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