If $y = \log_x x$, then the value of $\left( \frac{dy}{dx} \right)^2$ at $x = e^2$ is

The logarithm of $32\sqrt[5]{4}$ to the base $2\sqrt{2}$ is:

If $x = \log_a (bc)$, $y = \log_b (ca)$, and $z = \log_c (ab)$, then the value of $\frac{1}{1+x} + \frac{1}{1+y} + \frac{1}{1+z}$ is

The value of $6+\log_{\frac{3}{2}}\left(\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\dots}}}\right)$ is

$\log (9+3 \sqrt{2}(2+\sqrt{5})+4 \sqrt{5})=$

If $x = \log_3 5$ and $y = \log_{17} 25$,then which of the following is correct?