100 followers problem-2

Algebra Level 3

\[\large\log_{100}{\sqrt{2}}+\log_{100}{\sqrt[4]{2}}+\log_{100}{\sqrt[8]{2}}+\log_{100}{\sqrt[16]{2}}+\ldots\] Let \(A\) denote the value of series above. Find the value of \(\lfloor100A\rfloor\).

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