Question

solve the equation $2^{2x} = 32$. if no solution exists, enter none.

Explanation:

Step1: Express 32 as a power of 2

We know that \( 32 = 2^5 \), so the equation \( 2^{2x}=32 \) can be rewritten as \( 2^{2x}=2^5 \).

Step2: Set exponents equal (since bases are same)

For exponential equations with the same base (\( a^m = a^n \) implies \( m = n \) when \( a>0,a
eq1 \)), here \( a = 2 \), so we set the exponents equal: \( 2x=5 \).

Step3: Solve for x

Divide both sides of the equation \( 2x = 5 \) by 2. We get \( x=\frac{5}{2}=2.5 \).

Answer:

\( x = \frac{5}{2} \) (or \( 2.5 \))