All the logarithms

Calculus Level 5

\[f\left( x \right) =\ln { \left( \frac { 1 }{ \sqrt [ 2 ]{ \left| x \right| } } \right) } +\ln { \left( \frac { 1 }{ \sqrt [ 8 ]{ \left| x \right| } } \right) } +\ln { \left( \frac { 1 }{ \sqrt [ 24 ]{ \left| x \right| } } \right) } +\dots +\ln { \left( \frac { 1 }{ \sqrt [ n{ 2 }^{ n } ]{ \left| x \right| } } \right) }\]

Find the area bounded by \(f\left( x \right)\) and the \(x\)-axis. If this area can be expressed in the form \(\ln { A }\), find \(A\).

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