The number of solutions of the equation $\log _{4}(x-1)=\log _{2}(x-3)$ is

Let $m$ be the minimum possible value of $\log _3(3^{y_1}+3^{y_2}+3^{y_3})$,where $y_1, y_2, y_3$ are real numbers for which $y_1+y_2+y_3=9$. Let $M$ be the maximum possible value of $(\log _3 x_1+\log _3 x_2+\log _3 x_3)$,where $x_1, x_2, x_3$ are positive real numbers for which $x_1+x_2+x_3=9$. Then the value of $\log _2(m^3)+\log _3(M^2)$ is:

If $\log _{10} 7 = 0.8451$,then the position of the first significant figure of $7^{-20}$ is

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

Let $n$ be a positive integer such that $\log _2 \log _2 \log _2 \log _2 \log _2(n) < 0 < \log _2 \log _2 \log _2 \log _2(n)$. Let $l$ be the number of digits in the binary expansion of $n$. Then the minimum and the maximum possible values of $l$ are

The equivalent function of $\log {x^2}$ is