Question

Use a graphing utility to solve each equation. Express your answer rounded to two decimal places. $$ \ln (2 x)=-x+2 $$

Short answer

The solution is approximately x = 0.43.

Step-by-step solution

- Understand the Equation

The given equation is \ \( \ln(2x) = -x + 2. To solve for x, we'll use a graphing utility to find where the two functions \ \( y_1 = \ln(2x) \ \) and \ \ ( y_2 = -x + 2 \ \) intersect.

- Graph the Functions

Enter the functions \ \( y_1 = \ln(2x) \ \) and \ \( y_2 = -x + 2 \ \) into your graphing utility. Plot both graphs to visualize their intersections.

- Find the Intersection Points

Locate the point(s) where the graphs of \ \( y_1 \ \) and \ \( y_2 \ \) intersect. Note down the x-coordinates of these points, as these are the solutions to the original equation.

- Round the Solution

Round the x-coordinate(s) of the intersection point(s) to two decimal places as requested in the problem statement.

Key concepts

Understanding ln(2x)

The natural logarithm function, represented as `ln`, is the inverse of the exponential function when the base is `e` (approximately 2.718). When dealing with \(\text{ln}(2x)\), it means we are taking the natural logarithm of the product of 2 and x. This transformation stretches or compresses the natural logarithm function horizontally. Remember, the argument of the logarithm function must always be positive, meaning 2x > 0, or x > 0.

Graph Intersection

Graph intersection is determining where two functions meet on a graph. For the equation \( \text{ln}(2x) = -x + 2 \), we need to graph the functions \(y_1 = \text{ln}(2x)\) and \(y_2 = -x + 2\) and identify where they cross. These intersecting points' x-values are solutions to the equation. Graphs visually simplify the process of solving equations, especially when algebraic solutions might be complex or not straightforward.

Solving Equations with a Graphing Utility

Graphing utilities are helpful tools in solving equations. They allow you to plot functions easily and find their intersections. Here's a simple process:

• Input the functions you're analyzing
• Use the graph feature to visualize the functions
• Look for points where the graphs intersect; these are your solutions

This method is especially useful in finding solutions to equations where traditional methods might be more challenging to apply.

Rounding Solutions

Rounding is adjusting numbers to a desired precision. In this exercise, we round the intersection points' x-coordinates to two decimal places. This precision is common in many applications, providing a balance between accuracy and simplicity. To round to two decimal places: Look at the third decimal place; if it is 5 or more, round up the second decimal place. Otherwise, leave the second decimal place as it is.
Solve accurately then round the final answer.