Let's solve the equation step by step to find the correct value of [tex]\( x \)[/tex].
We start with the given equation:
[tex]\[ 216 = 6^{2x - 1} \][/tex]
To solve for [tex]\( x \)[/tex], we'll first express 216 as a power of 6. We know:
[tex]\[ 216 = 6^3 \][/tex]
because [tex]\( 6 \times 6 \times 6 = 216 \)[/tex].
So we can rewrite the equation as:
[tex]\[ 6^3 = 6^{2x - 1} \][/tex]
Since the bases are identical, we can equate the exponents:
[tex]\[ 3 = 2x - 1 \][/tex]
Next, solve for [tex]\( x \)[/tex]. To do this, we'll isolate [tex]\( x \)[/tex] on one side of the equation:
First, add 1 to both sides:
[tex]\[ 3 + 1 = 2x \][/tex]
[tex]\[ 4 = 2x \][/tex]
Now, divide both sides by 2:
[tex]\[ \frac{4}{2} = x \][/tex]
[tex]\[ x = 2 \][/tex]
Therefore, the solution to the equation [tex]\( 216 = 6^{2x - 1} \)[/tex] is:
[tex]\[ x = 2 \][/tex]
Among the provided options, the correct one is:
C. [tex]\( x = 2 \)[/tex]
