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

If ${\log _4}5 = a$ and ${\log _5}6 = b,$ then ${\log _3}2$ is equal to

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 integral solutions $x$ of $\log _{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2 x-3}\right)^2 \geq 0$ is

If ${a^x} = b,{b^y} = c,{c^z} = a,$ then value of $xyz$ is