Logs in an A.P.

Algebra Level 3

For \( x = \log _{ a }{ b }\) the three numbers \(\log _{ 10 }{ 2 }\) , \(\log _{ 10 }{ ({ 2 }^{ x } } -1)\) and \(\log _{ 10 }{ ({ 2 }^{ x } } +3)\) form an arithmetic progression.

What is the value of \((a+b)^{3}\) ?

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