To solve the equation:
[tex]\[ 324 = 4 \cdot 3^{2x} \][/tex]
we need to find the value of [tex]\( x \)[/tex] that satisfies this equation. Let's go through the steps to isolate [tex]\( x \)[/tex].
1. Isolate the exponential expression:
Divide both sides of the equation by 4 to simplify:
[tex]\[ \frac{324}{4} = 3^{2x} \][/tex]
Simplifying the left side:
[tex]\[ 81 = 3^{2x} \][/tex]
2. Express the left side with the same base:
Notice that [tex]\( 81 \)[/tex] can be expressed as a power of 3:
[tex]\[ 81 = 3^4 \][/tex]
Therefore, the equation becomes:
[tex]\[ 3^4 = 3^{2x} \][/tex]
3. Set the exponents equal to each other:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ 4 = 2x \][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides by 2:
[tex]\[ x = \frac{4}{2} \][/tex]
Simplifying the right side:
[tex]\[ x = 2 \][/tex]
So, the value of [tex]\( x \)[/tex] that satisfies the equation is [tex]\( x = 2 \)[/tex]. Therefore, the correct answer is:
C. [tex]\( x = 2 \)[/tex]
