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

The value of $(0.16)^{\log _{2.5}\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots . to \infty\right)}$ is equal to

The set of real values of $x$ for which ${\log _{0.2}}{{x + 2} \over x} \le 1$ is

The value of ${\log _3}\,4{\log _4}\,5{\log _5}\,6{\log _6}\,7{\log _7}\,8{\log _8}\,9$ is

Let $S$ be the sum of the digits of the number $15^2 \times 5^{18}$ in base $10$. Then,

The value of $\sqrt {(\log _{0.5}^24)} $ is