Express the solution of 2^(x+5) = 2 + 2^x in logarithmic form.

To explore why the correct answer can be rooted in logarithmic properties, let's first manipulate the equation given in the question:

Starting with ( 2^{(x+5)} = 2 + 2^{x} ), we can simplify this to better understand its logarithmic representation.

First, we rewrite the left side:

• ( 2^{(x+5)} ) can be expressed as ( 2^x \cdot 2^5 ) or ( 32 \cdot 2^x ).

The equation then transforms to:

• ( 32 \cdot 2^x = 2 + 2^x ).

Next, we can rewrite the equation by moving all terms involving ( 2^x ) to one side:

• ( 32 \cdot 2^x - 2^x = 2 ),

which simplifies to

• ( 31 \cdot 2^x = 2 ).

From here, we divide both sides by 31:

• ( 2^x = \frac{2}{31} ).

At this point, we can express ( x ) in logarithmic form. To isolate ( x \