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.

Explanation:

Step1: Isolate the logarithmic term

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

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

Using a calculator, \(e^{-\frac{1}{9}} \approx e^{-0.111\cdots} \approx 0.90\) (rounded to two decimal places). Also, check the domain: the domain of \(\ln x\) is \(x > 0\), and \(e^{-\frac{1}{9}}>0\), so it's valid.

Answer:

The exact solution is \(x = e^{-\frac{1}{9}}\) and the decimal approximation (to two decimal places) is \(x \approx 0.90\).