Find the Missing Exponent: Solving 32 = (1/2)^x

To solve this problem, let's follow these steps:

• Step 1: Recognize that 32 can be expressed as a power of 2.
• Step 2: Use negative exponents to equate $\left(\frac{1}{2}\right)^x$ to this power of 2.
• Step 3: Solve for the unknown exponent $x$.

Now, let's work through each step:
Step 1: The number 32 can be expressed as a power of 2. Specifically, $32 = 2^5$.
Step 2: The expression $\left(\frac{1}{2}\right)^x$ is equivalent to $\left(2^{-1}\right)^x = 2^{-x}$. To find $x$, equate: $2^5 = 2^{-x}$.
Step 3: Since the bases are equal, the exponents must be equal. Therefore, we have $5 = -x$. Solving for $x$, we multiply both sides by -1 and get $x = -5$.

Therefore, the solution to the problem is $x = -5$.