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

If ${\log _{10}}x = y,$ then ${\log _{1000}}{x^2} $ is equal to

${\log _4}18$ is

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

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 $(s)$ of the equation $log_7(2^x -1) + log_7(2^x -7) = 1$, is -