For the function $y = \log_a x$ to be defined,the base $a$ must be:

If the sum of the first $20$ terms of the series $\log _{7^{1/2}} x + \log _{7^{1/3}} x + \log _{7^{1/4}} x + \dots$ is $460$,then $x$ is equal to:

If $n = (2020)!$,then the value of $\frac{1}{\log _{2} n} + \frac{1}{\log _{3} n} + \frac{1}{\log _{4} n} + \ldots + \frac{1}{\log _{2020} n}$ is equal to:

The solution set of the equation $\left| {1 - {{\log }_{1/6}}x} \right| + \left| {{{\log }_2}x} \right| + 2 = \left| {3 - {{\log }_{1/6}}x + {{\log }_{1/2}}x} \right|$ is $\left[ {\frac{a}{b},a} \right]$,where $a, b \in N$. Then the value of $(a + b)$ is:

Let $\theta \in R$ such that $3 \sinh (2 \theta)=13-3 e^{2 \theta}$,then $\theta=$

If $x+\log _{15}\left(5+3^x\right)=x \log _{15} 5+\log _{15} 24$,then $x=\ldots .$.