If $\log_{a}b + \log_{b}c + \log_{c}a = 0$,where $a, b,$ and $c$ are positive real numbers different from $1$,then the value of $(\log_{a}b)^3 + (\log_{b}c)^3 + (\log_{c}a)^3$ is

Let $a, b, c$ be real numbers, each greater than $1$, such that $\frac{2}{3} \log _{b} a+\frac{3}{5} \log _{c} b+\frac{5}{2} \log _{a} c=3$. If the value of $b$ is $9$, then the value of $a$ must be

If ${\log _5}a \cdot {\log _a}x = 2$,then the value of $x$ is:

The number of solutions of the equation $\log _{(x+1)}(2 x^{2}+7 x+5)+\log _{(2 x+5)}(x+1)^{2}-4=0$ for $x > 0$ is:

Consider all natural numbers whose decimal expansion has only the even digits $0, 2, 4, 6, 8$. Suppose these are arranged in increasing order. If $a_n$ denotes the $n$-th number in this sequence,then $\frac{\lim_{n \rightarrow \infty} \log a_n}{\log n}$ equals

If $\log_{4}5 = a$ and $\log_{5}6 = b$,then $\log_{3}2$ is equal to