If ${\log _5}a.{\log _a}x = 2,$then $x$ is equal to

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 $........$.

Logarithm of $32\root 5 \of 4 $ to the base $2\sqrt 2 $ is

If ${a^2} + 4{b^2} = 12ab,$ then $\log (a + 2b)$ is

If ${\log _{10}}3 = 0.477$, the number of digits in ${3^{40}}$ is

The sum of all the natural numbers for which $log_{(4-x)}(x^2 -14x + 45)$ is defined is -