$A$ real root of the equation $\log_{4}\{\log_{2}(\sqrt{x + 8} - \sqrt{x})\} = 0$ 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:

The set of real values of $x$ that satisfy the inequality ${\log _{1/2}}({x^2} - 6x + 12) \ge - 2$ is:

Let $a, b, c$ be three distinct positive real numbers such that $(2a)^{\ln a} = (bc)^{\ln b}$ and $b^{\ln 2} = a^{\ln c}$. Then $6a + 5bc$ is equal to $........$.

If $x$ is a positive real number different from $1$ such that $\log _{a} x, \log _{b} x, \log _{c} x$ are in $AP$, then

Number of solution$(s)$ of the equation $\ln(1 + \sin^2 x) = 1 - \ln(5 + x^2)$ is -