Let $f(x) = {\log _{\cos x}}\cos ec\,x$ $= {{\log \cos ec\,x} \over {\log \cos x}}$ $\Rightarrow f'(x) = {{\log \cos x\,.\,\sin x\,.\,\left( { - \cos ec\,x\cot x - \log \cos ec\,x\,.\,{1 \over {\cos x}}\,.\, - \sin x} \right)} \over {{{(\log \cos x)}^2}}}$ at $x = {\pi \over 4}$ $f'\left( {{\pi \over 4}} \right) = {{ - \log \left( {{1 \over {\sqrt 2 }}} \right) + \log \sqrt 2 } \over {{{\left( {\log {1 \over {\sqrt 2 }}} \right)}^2}}} = {2 \over {\log \sqrt 2 }}$ $\therefore$ ${\log _e}2f'(x)$ at $x = {\pi \over 4} = 4$