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

The value of $\left(\frac{1}{\log _{3} 12}+\frac{1}{\log _{4} 12}\right)$ is

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

If $\log _{1/\sqrt{2}} \sin x > 0$ for $x \in [0, 4\pi]$,then the number of values of $x$ which are integral multiples of $\frac{\pi}{4}$ is:

The set of real values of $x$ for which $2^{\log_{\sqrt{2}}(x - 1)} > x + 5$ is

The product of all the solutions of the equation $x^{1 + \log_{10} x} = 100000x$ is