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, if ne\\( 3 + 9 ln x = 2 \\)\
\
rewrite the given equation without logarithms\
\\( x = e^{\frac{-1}{9}} \\)\
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.\
\\( \boldsymbol{\circ} \\) a. the solution set is \\( \langle \quad \
angle \\).\
\\( \quad \\) (type an exact answer. type your answer using exponential notation.)\
\\( \boldsymbol{\circ} \\) b. there are infinitely many solutions.\
\\( \boldsymbol{\circ} \\) 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:
\( x = e^{-\frac{1}{9}} \)

We check the domain: \( \ln x \) requires \( x > 0 \), and \( e^{-\frac{1}{9}} > 0 \) (since exponential functions are always positive), so it is valid.

Answer:

The solution set is \(\boldsymbol{e^{-\frac{1}{9}}}\) (or \(\boldsymbol{\frac{1}{e^{\frac{1}{9}}}}\)).