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Chapter 3: Problem 95

In Exercises \(93-102,\) solve each equation. $$2|\ln x|-6=0$$

Short Answer

Expert verified
The solutions to the equation \(2|\ln x|-6=0\) are \(x = e^3\) and \(x = e^{-3}\)

Step by step solution

01

Rewrite the equation

First, isolate the absolute term by moving 6 to the other side of the equation. Rewrite the equation as: \(2|\ln x|=6\)
02

Divide by 2

Now, divide both sides of the equation by 2. This results in \(|\ln x|=3\)
03

Split the absolute value

Next, split the absolute value into two equations, one for the positive value and another for the negative value of the log term. This gives us: \(\ln x = 3\) and \(\ln x = -3\)
04

Solve for x

We then exponentiate to solve for x in each equation (recall that the inverse of the natural log function is the exponential function). For the positive part we have \(x = e^3\) and for the negative part \(x = e^{-3}\)
05

Check the solutions

We find the solutions \(x = e^3\) and \(x = e^{-3}\). We have to verify these are in the domain of the original equation, i.e., they make sense in the context of \(\ln x\), which implies \(x > 0\). Both solutions are indeed positive, so they are valid

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Equation
When we come across an absolute value equation such as \(2|\ln x|-6=0\), it's much like encountering an

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