To find an approximate solution to the equation [tex]\(3^x = 2\)[/tex], we can follow these steps:
1. Take the natural logarithm of both sides:
ln([tex]\(3^x\)[/tex]) = ln(2)
2. Use the properties of logarithms (specifically, the power rule [tex]\( \ln(a^b) = b \cdot \ln(a) \)[/tex]):
[tex]\(x \cdot \ln(3) = \ln(2)\)[/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\( x = \frac{\ln(2)}{\ln(3)} \)[/tex]
Using numerical values from logarithms:
- [tex]\(\ln(2) \approx 0.6931\)[/tex]
- [tex]\(\ln(3) \approx 1.0986\)[/tex]
So,
[tex]\[
x \approx \frac{0.6931}{1.0986} \approx 0.6309297535714574
\][/tex]
When rounded to four decimal places, the solution is:
[tex]\[
x \approx 0.6309
\][/tex]
Thus, the correct choice is:
A. The solution set is [tex]\( \{ 0.6309 \} \)[/tex].
