If $\log_{5} a \cdot \log_{a} x = 2$,then $x$ is equal to

Evaluate the sum: $\sum\limits_{k = 1}^n \frac{1}{\log_{2^k}(a)}$

If $3^x = 4^{x-1}$,then $x = $

Below are four equations in $x$. Assume that $0 < r < 4$. Which of the following equations has the largest solution for $x$?

Find the solution of the equation $\log_{7}(\log_{5}(\sqrt{x^2 + x + 5})) = 0$.

The value of $x$ satisfying $\log _a x + \log _{\sqrt{a}} x + \log _{\sqrt[3]{a}} x + \dots + \log _{\sqrt[n]{a}} x = \frac{n(n+1)}{2}$ is given by the sum of $n$ terms. If the series is $\log _a x + \log _{a^{1/2}} x + \log _{a^{1/3}} x + \dots + \log _{a^{1/n}} x = S$,find $x$ for the given expression $\log _a x + \log _{\sqrt{a}} x + \dots + \log _{a^{1/n}} x = \frac{n(n+1)}{2}$. For the specific case provided: $\log _a x + \log _{a^{1/2}} x + \dots + \log _{a^{1/n}} x = \sum_{k=1}^{n} k \log_a x = \frac{n(n+1)}{2} \log_a x$. Given $\frac{n(n+1)}{2} \log_a x = \frac{a+1}{2}$,find $x$.