To solve the equation [tex]\(216 = 6^{2x - 1}\)[/tex], we can follow these steps:
1. Take the logarithm of both sides:
To isolate the exponent involving [tex]\(x\)[/tex], take the logarithm base [tex]\(6\)[/tex] of both sides of the equation.
[tex]\[
\log_6(216) = \log_6(6^{2x - 1})
\][/tex]
2. Apply the properties of logarithms:
Use the property of logarithms [tex]\(\log_b(a^c) = c \cdot \log_b(a)\)[/tex] on the right-hand side.
[tex]\[
\log_6(216) = (2x - 1) \cdot \log_6(6)
\][/tex]
3. Simplify the logarithm:
We know that [tex]\(\log_6(6) = 1\)[/tex], so the equation simplifies to:
[tex]\[
\log_6(216) = 2x - 1
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Rearrange the equation to solve for [tex]\(x\)[/tex].
[tex]\[
2x - 1 = \log_6(216)
\][/tex]
Add 1 to both sides:
[tex]\[
2x = \log_6(216) + 1
\][/tex]
Divide by 2:
[tex]\[
x = \frac{\log_6(216) + 1}{2}
\][/tex]
5. Evaluate the logarithm:
Now, evaluate the logarithm [tex]\(\log_6(216)\)[/tex]. The value is approximately 3.
6. Substitute the value:
[tex]\[
x = \frac{3 + 1}{2} = \frac{4}{2} = 2
\][/tex]
Thus, the correct solution to the equation is [tex]\(x = 2\)[/tex]. Therefore, the correct answer is:
A. [tex]\(x = 2\)[/tex]
