If $a^x = b^y = c^z = d^w$, the value of $x\left(\frac{1}{y} + \frac{1}{z} + \frac{1}{w}\right)$ is

The number of integers satisfying the inequality $\sqrt {{{\log }_3}(x) - 1} + \frac{{\frac{1}{2}{{\log }_3}({x^3})}}{{{{\log }_3}(\frac{1}{3})}} + 2 > 0$ is

$a^{m \log_a n} = ?$

The solution to the equation ${9^x} - {2^{x + 1/2}} = {2^{x + 3/2}} - {3^{2x - 1}}$ is:

The sum of all the natural numbers for which $\log_{(4-x)}(x^2 - 14x + 45)$ is defined is -

If $\log _{27}(\log _3 x) = \frac{1}{3}$,then the value of $x$ is