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 $\log_2 A$ .