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

If $n = (1999)!$,then $\sum\limits_{x = 1}^{1999} {{\log }_n x}$ is equal to

Number of solution$(s)$ of the equation $\ln(1 + \sin^2 x) = 1 - \ln(5 + x^2)$ is -

The expression $\log_{a} x$ is defined for $(a > 0, a \neq 1)$ when:

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 $abc =$

For a given number $\alpha > 1$,which of the following is the correct ascending order?