If ${\log _5}a.{\log _a}x = 2,$then $x$ is equal to

If ${\log _e}\left( {{{a + b} \over 2}} \right) = {1 \over 2}({\log _e}a + {\log _e}b)$, then relation between $a$ and $b$ will be

If $3^x=4^{x-1}$, then $x=$

$(A)$ $\frac{2 \log _3 2}{2 \log _3 2-1}$ $(B)$ $\frac{2}{2-\log _2 3}$ $(C)$ $\frac{1}{1-\log _4 3}$ $(D)$ $\frac{2 \log _2 3}{2 \log _2 3-1}$

The number of solution of ${\log _2}(x + 5) = 6 - x$ is

${\log _7}{\log _7}\sqrt {7(\sqrt {7\sqrt 7 } )} = $

Let $S$ be the sum of the digits of the number $15^2 \times 5^{18}$ in base $10$. Then,