In the equation 2^(x+5) = 2 + 2^x, what is the value of x?

To solve the equation (2^{(x+5)} = 2 + 2^x), we can start by rewriting the left-hand side. The term (2^{(x+5)}) can be expressed as (2^5 \cdot 2^x), which simplifies to (32 \cdot 2^x).

This gives us the new equation:

[

32 \cdot 2^x = 2 + 2^x

]

Next, we can isolate the (2^x) term. We do this by subtracting (2^x) from both sides:

[

32 \cdot 2^x - 2^x = 2

]

Factoring out (2^x) from the left-hand side results in:

[

(32 - 1) \cdot 2^x = 2

]

This simplifies to:

[

31 \cdot 2^x = 2

]

Next, we can divide both sides by 31:

[

2^x = \frac{2}{31}

]

To solve for (x), we take the logarithm