If $x = \log _2 \left( \sqrt {56 + \sqrt {56 + \sqrt {56 + \dots + \infty } } } \right)$,then:

The interval of $x$ in which the inequality $5^{\frac{1}{4}(\log_5 x)^2} \geq 5x^{\frac{1}{5}(\log_5 x)}$ holds is:

The number of solutions of the equation $\log _{(x+1)}(2 x^{2}+7 x+5)+\log _{(2 x+5)}(x+1)^{2}-4=0$ for $x > 0$ is:

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

Let $a, b, x$ be positive real numbers with $a \neq 1, x \neq 1, ab \neq 1$. Suppose $\log_{a} b = 10$,and $\frac{\log_{a} x \cdot \log_{x}(\frac{b}{a})}{\log_{x} b \cdot \log_{ab} x} = \frac{p}{q}$,where $p$ and $q$ are positive integers which are coprime. Then $p+q$ is

If ${\log _{10}}x = y$,then the value of ${\log _{1000}}{x^2}$ is