Question

solve the equation.
ln x + ln 3x = -1

select the correct choice and fill in any answer boxes in your choice below.

a. x =
(simplify your answer. round to four decimal places as needed.)

b. there is no solution.

Explanation:

Step1: Use logarithm property

Using the property \(\ln a+\ln b = \ln(ab)\), we have \(\ln x+\ln 3x=\ln(x\cdot3x)=\ln(3x^{2})\). So the equation becomes \(\ln(3x^{2})=- 1\).

Step2: Convert to exponential form

Recall that if \(\ln y = k\), then \(y = e^{k}\). So from \(\ln(3x^{2})=-1\), we get \(3x^{2}=e^{-1}=\frac{1}{e}\).

Step3: Solve for \(x^{2}\)

Divide both sides by 3: \(x^{2}=\frac{1}{3e}\).

Step4: Solve for \(x\)

Take the square root of both sides. Since \(x>0\) (because \(\ln x\) is defined for \(x > 0\)), we have \(x=\sqrt{\frac{1}{3e}}\). Calculate \(\sqrt{\frac{1}{3e}}\approx\sqrt{\frac{1}{3\times2.71828}}\approx\sqrt{\frac{1}{8.15484}}\approx\sqrt{0.1226}\approx0.3502\).

Answer:

A. \(x = 0.3502\)