Certainly, let's solve the equation [tex]\(9^{x-1} = 2\)[/tex].
1. Understand the equation:
[tex]\[
9^{x-1} = 2
\][/tex]
We want to solve for [tex]\(x\)[/tex] in this equation.
2. Introduce logarithms:
To solve for [tex]\(x\)[/tex], we can take the logarithm of both sides. For simplicity, we'll use the natural logarithm ([tex]\(\ln\)[/tex]).
[tex]\[
\ln(9^{x-1}) = \ln(2)
\][/tex]
3. Apply the power rule of logarithms:
The power rule states that [tex]\(\ln(a^b) = b\ln(a)\)[/tex]. Applying this rule helps in simplifying our equation.
[tex]\[
(x-1) \ln(9) = \ln(2)
\][/tex]
4. Isolate [tex]\(x-1\)[/tex]:
To isolate [tex]\(x-1\)[/tex], we need to divide both sides by [tex]\(\ln(9)\)[/tex].
[tex]\[
x-1 = \frac{\ln(2)}{\ln(9)}
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Adding 1 to both sides of the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{\ln(2)}{\ln(9)} + 1
\][/tex]
6. Calculate the numerical value:
Upon evaluating [tex]\(\frac{\ln(2)}{\ln(9)}\)[/tex], the result is approximately [tex]\(0.3154648767857287\)[/tex].
Therefore,
[tex]\[
x = 0.3154648767857287 + 1 \approx 1.3154648767857287
\][/tex]
7. Assess the options:
The options provided are:
- A. [tex]\(1\)[/tex]
- B. [tex]\(2\)[/tex]
- C. [tex]\(\frac{1}{2}\)[/tex]
- D. [tex]\(-\frac{1}{2}\)[/tex]
While the exact solution [tex]\(1.3154648767857287\)[/tex] does not match any of the given options exactly, it is closest to option A, which is [tex]\(1\)[/tex].
Thus, the most appropriate choice based on the closest value is:
[tex]\[
\boxed{1}
\][/tex]
