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What is the solution to this equation?
[tex]\[ 324 = 4(3)^{2x} \][/tex]

A. [tex]\( x = 1 \)[/tex]
B. [tex]\( x = 4 \)[/tex]
C. [tex]\( x = 2 \)[/tex]
D. [tex]\( x = 8 \)[/tex]

Sagot :

GinnyAnswer
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]
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