$0.5737373...... = $

Find the solution of the equation $9^x - 2^{x + 1/2} = 2^{x + 3/2} - 3^{2x - 1}$.

If ${a^x} = {b^y} = {(ab)^{xy}}$,then $x + y = $

If ${2^x} = {4^y} = {8^z}$ and $xyz = 288$,then find the value of $\frac{1}{{2x}} + \frac{1}{{4y}} + \frac{1}{{8z}}$.

Suppose $a, b, c$ are positive integers such that $2^a + 4^b + 8^c = 328$. Then,$\frac{a + 2b + 3c}{abc}$ is equal to

If for $x \in \left(0, \frac{\pi}{2}\right)$,$\log_{10} \sin x + \log_{10} \cos x = -1$ and $\log_{10}(\sin x + \cos x) = \frac{1}{2}(\log_{10} n - 1)$,$n > 0$,then the value of $n$ is equal to