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$
$\sum\limits_{n = 1}^n {{1 \over {{{\log }_{{2^n}}}(a)}}} = $
If ${\log _{0.3}}(x - 1) < {\log _{0.09}}(x - 1)$ then $x \ne 1$ lies in
Logarithm of $32\root 5 \of 4 $ to the base $2\sqrt 2 $ is
Which is the correct order for a given number $\alpha $in increasing order
If $a, b, c$ are distinct positive numbers, each different from $1$, such that $[{\log _b}a{\log _c}a - {\log _a}a] + [{\log _a}b{\log _c}b - {\log _b}b]$ $ + [{\log _a}c{\log _b}c - {\log _c}c] = 0,$ then $abc =$