We are asked to solve the equation $2^x + 2^x = 64$ for $x$.

1. Problem Description

We are asked to solve the equation $2^x + 2^x = 64$ for $x$.

2. Solution Steps

We have the equation $2^x + 2^x = 64$.

Since $2^x + 2^x = 2 \cdot 2^x$, we can rewrite the equation as:

$2 \cdot 2^x = 64$

We can simplify the left side further using the property $a^m \cdot a^n = a^{m+n}$:

$2^1 \cdot 2^x = 2^{1+x}$

So, we have $2^{x+1} = 64$.

We know that $64 = 2^6$. Thus, we have $2^{x+1} = 2^6$.

Since the bases are equal, we can equate the exponents:

$x+1 = 6$

Subtract 1 from both sides:

$x = 6 - 1$

$x = 5$

3. Final Answer

$x = 5$