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Chapter 3: Problem 52

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$2+3 \ln x=12$$

Short Answer

Expert verified
The approximate solution to the logarithmic equation \(2+3\ln x = 12\) is \(x \approx 28.033\).

Step by step solution

01

Isolate the logarithmic term

Subtract 2 from both sides of the equation to isolate the logarithmic term on one side. This will provide: \(3\ln x = 12 - 2\), which simplifies to \(3\ln x = 10\).
02

Get rid of the coefficient in front of the logarithm

Divide both sides of the equation by 3 to remove the coefficient in front of the logarithm. This results in \(\ln x = 10 / 3\), which simplifies to \(\ln x = 3.333\).
03

Convert to exponential form

Apply the property of logarithms that states the logarithm base \( e \) of a number is equal to \( y \) can be written as \( e^{y} = x \). This will transform the equation to exponential form: \(x = e^{3.333}\).
04

Solve for x

Use a calculator to compute \( e^{3.333} \) and get the approximate value of \( x \).
05

Check the solution

Ensure that the solution is within the domain of the original logarithmic function \(\ln x\), where \(x > 0\). If not, then the solution is extraneous and not valid.

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