Problem 95 - Algebra and Trigonometry

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: Isolate the Absolute Value

The equation is \(2|\ln x| - 6 = 0\). The first task is to isolate the absolute value on one side. This can be done by adding 6 to both sides and then dividing by 2. \(2|\ln x| = 6\) \(|\ln x| = 3\)

Step 2: Remove the Absolute Value

With absolute values, consider that either the positive or negative equivalent could equal to 3. That results in two separate equations to solve: \(\ln x = 3\) and \(\ln x = -3\).

Step 3: Solve for x

Next, solve each equation. Since the base of natural logarithm is e, take e to the power of each side. For the first equation: \(e^{ln x} = e^3\), resulting in \(x = e^3\). For the second equation: \(e^{ln x} = e^{-3}\), resulting in \(x = e^{-3}\). However, avoid solutions that would make the original log undefined or negative in the original equation. Thus, \(x = e^3\) and \(x = e^{-3}\) are the two solutions as \(e^3\) and \(e^{-3}\) are both positive.