Problem 95 - College Algebra Essentials

Question

Solve each equation. $$2|\ln x|-6=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = e^3\) and \(x = e^{-3}\)

Step 1: Simplify the Mathematical Equation

The first thing that needs to be done is to get the absolute value alone on one side of the equation. Do this by adding 6 to each side of the equation. This gives: \(2 | \ln x |=6\)

Step 2: Breaking Down the Absolute Value Term

Let's now remove the coefficient of absolute value by dividing the entire equation by 2. This results in \(| \ln x | = 3\). Now, let's break down the absolute value of lnx. Because an absolute value of a variable can be positive or negative, this gives two separate equations to solve: \( \ln x =3\) and \( \ln x =-3 \)

Step 3: Solve the Two Equations

We now have two equations to solve for x. We can do so by exponentiating both sides of each equation. This will give two possible solutions for x. From the equation \( \ln x =3\), the solution yields \(x = e^3\). And from the equation \( \ln x =-3\), the solution yields \(x = e^{-3}\)