Problem 78 - College Algebra with Modeling and Visualization

Question

Solve the equation graphically. Express any solutions to the nearest thousandth. $$ \log _{3}\left(1+x^{2}+2 x^{4}\right)=4 $$

Step-by-Step Solution

Answer

Solutions are approximately \( x \approx -2.371 \) and \( x \approx 2.371 \).

Step 1: Rewrite the Equation

Start by rewriting the equation in a more standard form. The given equation is \( \log_{3}(1+x^2+2x^4) = 4 \). This can be rewritten using the property of logarithms as \( 1+x^2+2x^4 = 3^4 \).

Step 2: Calculate the Right Side

Calculate the numerical value of \( 3^4 \) to simplify the expression. \[ 3^4 = 81 \]

Step 3: Set Up the Graphing Equation

Set the two expressions equal to each other to form an equation to graph. The equation becomes \( 1 + x^2 + 2x^4 = 81 \). Simplify it to \( 2x^4 + x^2 + 1 = 81 \) or \( 2x^4 + x^2 - 80 = 0 \).

Step 4: Graph the Function

Graph the function \( y = 2x^4 + x^2 - 80 \) using a graphing calculator or software. Look for the points where the graph intersects the line \( y = 0 \). These intersections represent the real solutions to the equation.

Step 5: Identify the Solutions

From the graph, identify the x-values where the function crosses the x-axis (i.e., \( y = 0 \)). These x-values are the solutions to the equation.

Step 6: Approximate the Solutions to the Nearest Thousandth

Using the graph, approximate the x-values where the function meets the x-axis to three decimal places. These approximations need to be calculated carefully either visually or with the aid of the calculator's 'zero' function.

Key Concepts

Logarithmic equations

Logarithmic equations involve variables inside a logarithm, requiring specific techniques to solve, especially when utilizing graphical methods.
When dealing with a logarithmic equation like \( \log_{3}(1+x^2+2x^4) = 4 \), the first step is to convert it to a different form that may be easier to handle. By leveraging the definition of a logarithm, this equation can be rewritten as an exponentiation problem: the contents of the logarithm equal the base of the logarithm raised to the other side of the equation. This transformation gives us \( 1 + x^2 + 2x^4 = 81 \), making the equation more approachable for solving graphically.
The principle here is that solving logarithmic equations often involves changing them into exponential equations. This practice not only simplifies solving but also exposes the polynomial or other functions hidden inside the logarithm. Once transformed, traditional solving techniques, including graphical analysis, can be applied.

Polynomial equations

Polynomial equations are expressed in the form \( a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0 = 0 \), where variables are raised to whole number powers.
In our context, after converting the logarithmic form to a polynomial form \( 2x^4 + x^2 - 80 = 0 \), we need to find the roots of this polynomial.
Graphically, the roots of the polynomial are the x-values at which the polynomial equals zero. In simpler terms, these are the points where the graph of the polynomial will intersect the x-axis.

• Identify structure: The degree of the polynomial, here 4, indicates a quartic equation which might have up to four real or complex solutions.
• Factoring: While tertiary methods like factoring can be used, they aren’t always straightforward with higher-powered equations as seen here.

This is where graphing calculators become a useful tool to easily visualize where solutions lie.

Using graphing calculators

A graphing calculator is an essential tool for visualizing and solving equations, making it particularly useful for complex logarithmic and polynomial equations.
To solve our equation graphically using a calculator, follow these steps:

• Input the function: Enter \( y = 2x^4 + x^2 - 80 \) into the calculator.
• Graph the polynomial: The graph will show a curve that might go above and below the x-axis.
• Find intersections with x-axis: Look for points where the graph intersects the x-axis. These are potential solutions.
• Use zero and trace functions: To ensure accuracy, utilize the calculator’s zero or root functions to pinpoint exact solutions. This feature calculates approximations, often to the nearest thousandth.

Graphing calculators streamline this process, transforming complex algebraic manipulations into visual problem-solving. Recognizing and confirming solutions become more intuitive and accessible with graphing technology.