Question

solve the following logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expression. give the exact answer. then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution.
3 + 9 ln x = 2
rewrite the given equation without logarithms.
solve the equation. what is the exact solution? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is \boxed{e^{-\frac{1}{9}}}. (type an exact answer. type your answer using exponential notation.)
b. there are infinitely many solutions.
c. there is no solution.
what is the decimal approximation to the solution? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is \boxed{}. (type an integer or decimal rounded to two decimal places as needed.)
b. there are infinitely many solutions.
c. there is no solution.

Explanation:

Step1: Isolate the logarithmic term

Start with the equation \( 3 + 9\ln x = 2 \). Subtract 3 from both sides:
\( 9\ln x = 2 - 3 \)
\( 9\ln x = -1 \)

Step2: Solve for \(\ln x\)

Divide both sides by 9:
\( \ln x = \frac{-1}{9} \)

Step3: Convert to exponential form

Recall that \( \ln x = y \) is equivalent to \( x = e^y \). So:
\( x = e^{-\frac{1}{9}} \)

Step4: Approximate the decimal value

Calculate \( e^{-\frac{1}{9}} \). Using a calculator, \( e^{-1/9} \approx e^{-0.1111} \approx 0.8958 \) (rounded to four decimal places, or 0.90 to two decimal places).

Answer:

Exact solution: \( \boldsymbol{e^{-\frac{1}{9}}} \)
Decimal approximation (to two decimal places): \( \boldsymbol{0.90} \)