For $y = {\log _a}x$ to be defined $'a'$ must be
Any positive real number
Any number
$ \ge e$
Any positive real number $ \ne 1$
The interval of $x$ in which the inequality ${5^{(1/4)(\log _5^2x)}}\, \geqslant \,5{x^{(1/5)(\log _5^x)}}$
The number ${\log _2}7$ is
Let $S$ be the sum of the digits of the number $15^2 \times 5^{18}$ in base $10$. Then,
The number of solution pairs $(x, y)$ of the simultaneous equations $\log _{1 / 3}(x+y)+\log _3(x-y)=2$ $2^{y^2}=512^{x+1}$ is
Let $\log _a b=4, \log _c d=2$, where $a, b, c, d$ are natural numbers. Given that $b-d=7$, the value of $c-a$ is