If ${\log _4}5 = a$ and ${\log _5}6 = b,$ then find the value of ${\log _3}2$.

Which one of the following observations is correct for the features of the logarithm function to any base $b > 1$?

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 $x$.

If $\alpha = \log_e(2+\sqrt{3})$,then $\frac{\cosh \alpha}{1-\tanh \alpha} + \frac{\sinh \alpha}{1-\coth \alpha} = $

If $n = 1000!$,then $\frac{1}{\log_2 n} + \frac{1}{\log_3 n} + ... + \frac{1}{\log_{1000} n} = ......$

$\log (\cosh 3 + \sinh 3) + \log (\cosh 3 - \sinh 3) = $