Let's solve the given equation step by step:
The equation given is:
[tex]\[ 324 = 4 \cdot 3^{2x} \][/tex]
1. Isolate the exponential term:
We start by dividing both sides of the equation by 4 to isolate the exponential term.
[tex]\[ \frac{324}{4} = 3^{2x} \][/tex]
[tex]\[ 81 = 3^{2x} \][/tex]
2. Express the number as a power of the base:
Notice that 81 can be expressed as a power of 3:
[tex]\[ 81 = 3^4 \][/tex]
So, 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]:
To find [tex]\( x \)[/tex], divide both sides by 2:
[tex]\[ x = \frac{4}{2} \][/tex]
[tex]\[ x = 2 \][/tex]
So, the solution to the equation [tex]\( 324 = 4 \cdot 3^{2x} \)[/tex] is:
[tex]\[ x = 2 \][/tex]
Therefore, the correct answer is not listed among the given options. The given options are incorrect. The correct value of [tex]\( x \)[/tex] is 2.
