Problem 56 - Algebra and Trigonometry

Question

Use a graphing device to find all solutions of the equation, correct to two decimal places. $$ \log x=x^{2}-2 $$

Step-by-Step Solution

Answer

Using a graphing device, the solutions to the equation are approximately \( x = 1.54 \) and \( x = e \).

Step 1: Understand the Equation

The equation we need to solve is \( \log x = x^2 - 2 \). The left side is a logarithmic function, and the right side is a quadratic function. Our task is to find where these two functions intersect.

Step 2: Set Up the Functions

Identify the two functions: \( f(x) = \log x \) and \( g(x) = x^2 - 2 \). We will find the intersection points of these functions using a graphing calculator or software.

Step 3: Graph the Functions

Use a graphing device to plot both functions on the same set of axes. The \( y\)-axis represents the values of both \( f(x) \) and \( g(x) \), while the \( x\)-axis represents the values of \( x \).

Step 4: Identify Intersection Points

Using the graph on the device, look for points where the two curves intersect. These points represent the solutions to the equation \( \log x = x^2 - 2 \).

Step 5: Determine Accurate Solutions

Zoom in on the intersection points to accurately identify the \( x \)-values. From the graph, find the \( x \)-coordinates to two decimal places.

Key Concepts

Logarithmic Function

A logarithmic function is a mathematical function that involves the logarithm of a variable. In this particular exercise, we deal with the function expressed as \( f(x) = \log x \). The logarithm is the inverse operation of exponentiation, meaning it's used to determine the power to which a base number is raised to obtain another number.
Common bases for logarithms are 10 (common logarithm) and \( e \), Euler's number (the natural logarithm). For most graphing applications, the base used in a calculator is typically the natural base \( e \), unless specified otherwise.
Understanding how the \( \log x \) function behaves is crucial. It is defined only for positive values of \( x \), and as \( x \) becomes larger, \( \log x \) increases at a decreasing rate. At \( x = 1 \), \( \log x \) equals 0, signifying that it passes through this point on the graph with the y-axis.

Quadratic Function

A quadratic function is one of the simplest types of polynomial functions, and it is characterized by the general form \( g(x) = ax^2 + bx + c \). For our exercise, the quadratic function is given as \( g(x) = x^2 - 2 \).
This particular quadratic equation does not have a linear term, so it's a simple parabola centered horizontally along the y-axis. The constant term, \(-2\), shifts the graph downward by two units.
Quadratic functions are known for their symmetry and can open upwards or downwards depending on the coefficient of the \( x^2 \) term. For \( x^2 - 2 \), the graph opens upwards because the coefficient of \( x^2 \) is positive. Understanding its characteristics helps in determining where it might intersect with the logarithmic function on a graph.

Graphing Calculator

A graphing calculator is an essential tool for visualizing mathematical functions. It helps you graph functions and find their intersections, providing a visual representation that makes it easier to connect abstract algebraic concepts with visual data.
To solve the equation \( \log x = x^2 - 2 \) using a graphing device, input both functions, \( f(x) = \log x \) and \( g(x) = x^2 - 2 \), into the calculator. This will plot both curves on the same grid, showing their behavior over different values of \( x \).

• Ensure both functions are entered correctly, and check the range of \( x \) values.
• Use features like \( x \)-min/max and \( y \)-min/max to adjust the window for better visualization.

By zooming in on intersection points, you can read precise values for \( x \), allowing you to solve the equation accurately.

Intersection Points

Intersection points on a graph are where two functions meet. In the context of solving equations graphically, these points are crucial as they represent solutions to the equation. For the exercise \( \log x = x^2 - 2 \), the intersection points of the functions \( f(x) = \log x \) and \( g(x) = x^2 - 2 \) are what we're after.
When using a graphing calculator, once both functions are plotted:

• Look for points where both curves cross each other.
• Zoom in to these points to determine precise x-coordinates.

Each intersection point on the x-axis corresponds to a value of \( x \) where \( \log x \) equals \( x^2 - 2 \). Calculate these solutions to two decimal places for accuracy. This graphical method offers an insightful way to solve complex equations that might be hard to handle algebraically.