If ${\log _{10}}3 = 0.477$,then the number of digits in ${3^{40}}$ is

If $n = 1000!$,then $\frac{1}{\log_2 n} + \frac{1}{\log_3 n} + ... + \frac{1}{\log_{1000} n} = ......$

If $\log _{0.3}(x-1) < \log _{0.09}(x-1),$ then $x$ lies in the interval

Given that $a, b \in \{0, 1, 2, \ldots, 9\}$ with $a+b \neq 0$ and that $\left(a+\frac{b}{10}\right)^x = \left(\frac{a}{10}+\frac{b}{100}\right)^y = 1000$. Then, $\frac{1}{x} - \frac{1}{y}$ is equal to

The value of $\log_{\frac{1}{8}\csc^2\frac{\pi}{8}} \sin^2\frac{3\pi}{8}$ is equal to

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