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Chapter 3: Problem 25

Use the One-to-One Property to solve the equation for \(x .\) $$\left(\frac{1}{2}\right)^{x}=32$$

Short Answer

Expert verified
The solution to the equation \((1/2)^x = 32\) is \(x = -5\).

Step by step solution

01

Rewrite the Equation with the Same Base

To rewrite the equation with the same base, we need to find a base that both 1/2 and 32 can be expressed as powers of. In this case, the base is 2. So first, note that \((1/2)^x\) can be written as \(2^{-x}\). Second, 32 is 2 to the power of 5 or \(2^{5}\). So, our equation becomes \(2^{-x} = 2^{5}\).
02

Apply the One-to-One Property

Once the equation has been rewritten with the same base, we can apply the One-to-One Property. This means, if the bases are equal, then the exponents also have to be equal. So, we set -x equal to 5, or \(-x =5\).
03

Solve for x

Finally, we solve for x by multiplying both sides of the equation by -1 to get x on its own. So, \(x = -5\).

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