Problem 67 - Precalculus with Limits

Question

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. \(3-\ln x=0\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 20.086\)

Step 1: Rearrange the Equation

In the given equation \(3 - \ln{x} = 0\), the first step is to rearrange the equation to get \( \ln{x} \) alone. Adding \(\ln{x}\) to both sides of the equation gives \( \ln{x} = 3 \)

Step 2: Graphing and Solving

Using a graphing utility (like a graphing calculator or online graphic tool), graph the function \(y = \ln{x}\). The point where the curve intersects \( y=3 \) will give the approximate value for x.

Step 3: Algebraic Verification

To verify the solution algebraically, take the exponent of both sides of the equation \(ln(x) = 3\) in base \(e\) to get rid of the log, resulting in \(x = e^{3}\). Calculate the value of \(e^{3}\) approximated to three decimal places. The approximately value of \(e^3\) is 20.086

Key Concepts

Logarithmic Functions

Logarithmic functions are an essential concept in mathematics, often appearing in a variety of mathematical contexts, including growth processes and scientific computations. A logarithm, represented as \( \ln(x) \) when using the natural logarithm, essentially answers the question: "To what power must we raise a base, in this case \( e \), to obtain the value \( x \)."

Here, the equation \( 3 - \ln x = 0 \) involves the natural logarithm \( \ln x \). To solve it, we isolate \( \ln x \) by moving it to one side, resulting in \( \ln x = 3 \). This equation informs us that we need to find a number, \( x \), such that \( \ln(x) = 3 \). By applying the concept of logarithms, we infer that \( e^3 \) is the number we're looking for.

• Natural logarithms have a base of \( e \), an irrational number approximately equal to 2.718.
• They are the inverse of the exponential function of base \( e \).
• When graphed, the natural logarithm function \( \ln(x) \) has a continuous curve.

Graphing Utilities

Graphing utilities are indispensable tools for visualizing functions and solving equations graphically. They can take various forms, such as graphing calculators, online graphing tools, or software programs. These tools allow users to plot functions, zoom in for details, and find intersections or points of interest on a graph.

In our problem, after finding \( \ln x = 3 \), we use a graphing utility to visualize the functions \( y = \ln x \) and \( y = 3 \). By plotting these on a common coordinate system, the intersection point of the natural logarithm curve with the horizontal line at \( y = 3 \) gives the approximate value of \( x \).

• These tools allow you to see the shape of the function \( \ln(x) \) across different values of \( x \).
• They can pinpoint where \( \ln(x) \) achieves the value 3, aligning with its horizontal line.
• Graphing utilities are vital for learning visually and understanding the behavior of complex functions.

Exponential Functions

Exponential functions are the inverse of logarithmic functions and play a crucial role in algebra and calculus. Given the relationship between exponentials and logarithms, an equation like \( \ln x = 3 \) can be transformed into an exponential equation.

By exponentiating both sides, we convert the logarithmic equation into \( x = e^3 \). This transformation allows us to solve for \( x \) directly, using the properties of exponentials. The constant \( e \) is significant because it is the base of natural logarithms, making it highly relevant in continuous growth contexts such as population growth or compound interest.

• An exponential function with base \( e \), represented as \( e^x \), grows rapidly.
• Solving \( x = e^3 \) helps us verify that using logarithms and exponentials leads to the same answer.
• These functions provide intuitive understanding in modeling real-world phenomena where growth patterns emerge naturally.