The expression $\log_{a} x$ is defined for $(a > 0, a \neq 1)$ when:

If $x, y, z \in R^+$ are such that $z > y > x > 1$,$\log_{y}x + \log_{x}y = \frac{5}{2}$ and $\log_{z}y + \log_{y}z = \frac{10}{3}$,then $\log_{x}z$ is equal to

The sum of the roots of the equation $x+1-2 \log _{2}\left(3+2^{x}\right)+2 \log _{4}\left(10-2^{-x}\right)=0$ is:

If $a^x = b^y = c^z = d^w$,then the value of $x\left(\frac{1}{y} + \frac{1}{z} + \frac{1}{w}\right)$ is

Evaluate the sum: $\sum\limits_{k = 1}^n \frac{1}{\log_{2^k}(a)}$

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: