Questions: Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents. 2^(3-x) = 1/4 The solution set is B.

Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
2^(3-x) = 1/4

The solution set is B.

Transcript text: Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents. \[ 2^{3-x}=\frac{1}{4} \] The solution set is $\square$ B.

Solution

Solution Steps

To solve the exponential equation $2^{3-x} = \frac{1}{4}$, we need to express both sides of the equation with the same base. Notice that $\frac{1}{4}$ can be written as $2^{-2}$. Once both sides are expressed as powers of the same base, we can equate the exponents and solve for $x$.

Solution Approach

Rewrite $\frac{1}{4}$ as $2^{-2}$.
Set the exponents equal to each other: $3 - x = -2$.
Solve the resulting linear equation for $x$.

Step 1: Rewrite the Equation

We start with the equation: \[ 2^{3-x} = \frac{1}{4} \] We can express $\frac{1}{4}$ as a power of $2$: \[ \frac{1}{4} = 2^{-2} \] Thus, the equation becomes: \[ 2^{3-x} = 2^{-2} \]

Step 2: Equate the Exponents

Since the bases are the same, we can equate the exponents: \[ 3 - x = -2 \]

Step 3: Solve for $x$

To solve for $x$, we rearrange the equation: \[ -x = -2 - 3 \] \[ -x = -5 \] Multiplying both sides by $-1$ gives: \[ x = 5 \]