Multiplication of Power Tower

Algebra Level 3

$\large \left( 1+{ 2 }^{ { 2 }^{ 0 } } \right) \left( 1+{ 2 }^{ { 2 }^{ 1 } } \right) \left( 1+{ 2 }^{ { 2 }^{ 2 } } \right) \cdots\left( 1+{ 2 }^{ { 2 }^{ 2015 } } \right)$

If the expression above can be expressed as $$2^A - 1$$, find the value of $$\log_2 A$$.

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