$7 \log \left( \frac{16}{15} \right) + 5 \log \left( \frac{25}{24} \right) + 3 \log \left( \frac{81}{80} \right)$ is equal to

If $\frac{1}{\log_3 \pi} + \frac{1}{\log_4 \pi} > x$,then $x$ can be:

Let $a, b, x$ be positive real numbers with $a \neq 1, x \neq 1, ab \neq 1$. Suppose $\log_{a} b = 10$,and $\frac{\log_{a} x \cdot \log_{x}(\frac{b}{a})}{\log_{x} b \cdot \log_{ab} x} = \frac{p}{q}$,where $p$ and $q$ are positive integers which are coprime. Then $p+q$ is

$\sinh[\log(2+\sqrt{5})] + \cosh[\log(2+\sqrt{3})] = ?$

Let $k>0$ and $t=\operatorname{sech}^{-1}\left(\frac{1}{2}\right)-\operatorname{cosech}^{-1}\left(\frac{3}{k}\right)$. If $3 e^t=2+\sqrt{3}$,then $k=$

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 $x$.