Problem 78 - Precalculus with Limits

Question

Solve the equation algebraically. Round your result to three decimal places. Verify your answer using a graphing utility. \(\frac{1-\ln x}{x^{2}}=0\)

Step-by-Step Solution

Answer

The solution to the equation \(\frac{1 - \ln{x}}{x^2} = 0\) is \(x = e\) or approximately \(x = 2.718\)

Step 1: Isolate the logarithmic expression

First, let's solve \(1 - \ln{x} = 0\) since the equation \(\frac{1 - \ln{x}}{x^2} = 0\) implies that the numerator of the fraction is equal to zero. To do this, we move \(\ln{x}\) to the other side of the equation: \(1 = \ln{x}\)

Step 2: Convert the logarithm to exponential form

The natural log function \(\ln{x}\) is also known as logarithm with base \(e\). Therefore, we can rewrite \(1 = \ln{x}\) as \(e^1 = x\). Given that \(e^1\) is just \(e\), we can simplify this to \(x = e\) as a solution.

Step 3: Verify the solution.

Substitute \(x = e\) into the original equation to verify if it indeed solves the equation: \(\frac{1 - \ln{e}}{e^2} = \frac{1 - 1}{e^2} = 0\). So, \(x = e\) is a valid solution.

Step 4: Check the graph

To further verify the solution, graph the function and observe whether \(x = e\) is indeed a root. If it is, there will be a point on the graph of our function at \(x = e\) that touches the x-axis.

Key Concepts

Natural Logarithm

A natural logarithm is a logarithm with a special base, known as "e." The constant "e" is approximately equal to 2.71828 and is a central element in mathematics, closely related to growth and exponential functions. The natural logarithm of a number is the power to which you must raise "e" to obtain that number. Thus, if we say \( \ln{x} = y \), it means \( e^y = x \). The natural log is prevalent in many mathematical fields due to its unique properties that make it easier to handle compared to other logarithmic bases. Understanding natural logs provides the foundation for solving equations like \( 1 = \ln{x} \), which leads us directly to the solution of \( x = e \) by converting the logarithmic expression into exponential form.

Exponential Functions

Exponential functions involve the mathematical constant "e" and can be written in the form \( f(x) = a \cdot e^{bx} \), where \( a \) and \( b \) are constants. These functions display continuous growth or decay, which is why they're often used to model real-world situations like population growth, radioactive decay, and interest calculations. The function \( e^x \) is particularly important because it manifests the simplest form of exponential growth. For example, solving \( \ln{x} = 1 \) and rewriting it as \( e^1 = x \) shows us that the point where the natural log equals 1 corresponds to \( x = e \), demonstrating the power of converting logarithmic terms into an exponential form to find solutions easily.

Graphical Verification

Graphical verification is a crucial step in solving equations, offering visual confirmation of the solutions. When an equation is solved algebraically, plotting the function on a graph can help verify the accuracy by checking if the solutions correspond to points where the function intercepts the x-axis (roots). In this exercise, once we find \( x = e \) as a solution, plotting the original equation \( \frac{1 - \ln{x}}{x^2} \) allows us to see if there's a clear intersection at \( x = e \). This graphical intersection confirms that we have correctly identified the root of the equation, ensuring our solution is not merely theoretical but also visually confirmed.

Roots of Equations

Roots of an equation are values that make the equation true, essentially where the equation "comes down to" zero. In mathematical notation, solving for roots means finding \( x \) such that \( f(x) = 0 \). For the given equation \( \frac{1 - \ln{x}}{x^2} = 0 \), setting the numerator to zero \( 1 - \ln{x} = 0 \) effectively finds the roots. After simplifying, this leads us to \( x = e \), a root where the left side of the equation equals zero. Understanding how to isolate and find these roots is crucial for solving algebraic equations, and it allows us to analyze more complex mathematical models easily. For any function graphed on a coordinate plane, the roots are where the plot touches or crosses the x-axis.