Question

Use a graphing utility to solve each equation. Express your answer rounded to two decimal places. $$ \ln x=x^{3}-1 $$

Short answer

The solutions are approximately x = 1.18 and x = 1.96.

Step-by-step solution

Understand the Equation

The given equation is \( \ln x = x^{3} - 1 \). This equation involves both a natural logarithm and a cubic expression.

Set Up the Graph

Graph both functions \( y = \ln x \) and \( y = x^{3} - 1 \) using a graphing utility. We will look for the points where these two curves intersect.

Plot the Functions

First, plot the function \( y = \ln x \) which is defined for \( x > 0 \). Then, plot the function \( y = x^{3} - 1 \) which is defined for all real numbers.

Find the Intersection Points

Observe the graph to find the intersection points of the two curves. These points satisfy the equation \( \ln x = x^{3} - 1 \)

Read the Solutions

Using the graphing utility, find the x-values of the intersection points. Round each solution to two decimal places.

Key concepts

Natural Logarithm

The natural logarithm, denoted as \(ln x\), is a fundamental concept in mathematics. It is the logarithm to the base \(e\) (where \(e\) is approximately 2.71828). The natural logarithm of a number \(x\) answers the question: 'To what power must \(e\) be raised to get \(x\)?'. For example, if \(e^2 = 7.389\), then \(ln(7.389) = 2\).
The natural logarithm function is only defined for positive values of \(x\). This means that in the given problem, \(ln x\) will be graphed only where \(x > 0\). This is important to remember as it helps us understand what parts of the graph are relevant. The function increases slowly and is always positive for positive \(x\), which shapes how we interpret its intersection with other functions.

Cubic Function

A cubic function is a polynomial function of degree three. It is generally expressed in the form \(y = ax^3 + bx^2 + cx + d\). In our problem, the cubic function given is \(y = x^3 - 1\). This function will cover all real numbers and is defined everywhere.
The shape of a cubic function includes one or more turns, creating an 'S' shape on the graph. It can increase or decrease depending on the values of \(x \). The behavior of this function is critical as it allows us to find where it intersects with the natural logarithm function. This intersection provides solutions to the given equation. This specific cubic function \(x^3 - 1\) shifts the graph of \(x^3\) downward by 1 on the y-axis, which can help us visually locate the intersection points.

Intersection Points

Finding intersection points of two functions on a graph means identifying where the functions have the same value at the same \(x \) coordinate. In our equation \(ln x = x^3 - 1\), these intersections represent the values of \(x \) that satisfy this equation.
To locate these intersection points:

• Graph both \(y = ln x\) and \(y = x^3 - 1\).
• Look for points where the curves cross each other.
• Use the graphing utility to read off the \(x \) values of these points.

This task is easier with a graphing utility because it provides visual confirmation. The \(x \) values at these intersections, rounded to two decimal places, give us the solutions to our equation. These intersection points are not only valuable mathematically but also provide a deeper insight into how different kinds of functions interact with each other within the same coordinate system.

Graphing Functions

Graphing functions is a useful method for visually analyzing equations and systems. A graphing utility (such as a graphing calculator or software) allows us to create visual representations of the functions \(y = ln x\) and \(y = x^3 - 1\).
To graph these functions:

• Input the functions into the graphing utility.
• Ensure the domain and range fit within the viewing window. This often requires adjusting zoom or pan settings.
• Identify and label the intersection points accurately where the curves cross.

Graphing utility tools provide not only visual aid but allow precise calculations of values where functions intersect. For this problem, such tools are particularly helpful for verifying the x-values obtained when \(ln x\) and \(x^3 - 1\) equal each other. Seeing these functions overlap on a graph helps students appreciate the solution-finding process.