Problem 61 - Intermediate Algebra

Question

What is the solution for \(3^{x}=5\) ? Do you agree that it is between 1 and 2 because \(3^{1}=3\) and \(3^{2}=9\) ? Now graph \(f(x)=3^{x}-5\) and use the ZOOM and TRACE features of your graphing calculator to find an approximation, to the nearest hundredth, for the \(x\) intercept. You should get an answer of \(1.46\), to the nearest hundredth. Do you see that this is an approximation for the solution of \(3^{x}=5\) ? Try it; raise 3 to the \(1.46\) power. Find an approximate solution, to the nearest hundredth, for each of the following equations by graphing the appropriate function and finding the \(x\) intercept. (a) \(2^{x}=19\) (b) \(3^{x}=50\) (c) \(4^{x}=47\) (d) \(5^{x}=120\) (e) \(2^{x}=1500\) (f) \(3^{x-1}=34\)

Step-by-Step Solution

Answer

(a) 4.25, (b) 3.56, (c) 2.57, (d) 2.89, (e) 10.55, (f) 4.52.

Step 1: Understand the function

For each equation, rearrange into a function where you're finding when the equation equals zero. For instance, for (a) use \(f(x) = 2^x - 19\). The solution will be where this function crosses the x-axis (i.e., \(f(x) = 0\)).

Step 2: Graph the function

Input each function into a graphing calculator to plot the equation. For instance, use \( f(x) = 2^x - 19 \) to create a visual representation.

Step 3: Zoom in on the graph

Utilize the ZOOM feature on your calculator to get a clearer view around the x-axis intercept point. This will give you a more precise visual on where the line crosses the x-axis.

Step 4: Trace the x-axis

Use the TRACE function on your graphing calculator to move along the graph and find the point where the function crosses the x-axis. Record the x-coordinate at this point.

Step 5: Approximate the solution

Find the x-intercept to the nearest hundredth by using the TRACE feature. This value represents the approximate solution to the equation.

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool for visualizing and solving complex mathematical equations, such as exponential equations. It allows students to input functions and view their graphs, making it easier to identify important components like the x-intercept. For example, when working with an equation like \( 3^x = 5 \), a graphing calculator simplifies the process by graphing \( f(x) = 3^x - 5 \).

Graphing calculators often come with features like ZOOM and TRACE, which can be incredibly helpful for obtaining precise values and approximations. By taking advantage of these features, students can enhance their understanding of the behavior and properties of functions they are studying. This technology allows for a hands-on approach to learning that can deepen comprehension and make abstract concepts more tangible.

x-intercept

The x-intercept of a graph is a fundamental point where the graph of a function crosses the x-axis. Mathematically, it's the value of \( x \) when the function \( f(x) \) equals zero. When solving the equation \( 3^x = 5 \), for instance, it converts to finding the x-intercept of the function \( f(x) = 3^x - 5 \).

Identifying the x-intercept is crucial since it represents the solution to the equation. In a graphing calculator, by tracing and zooming in, you can determine the point where the graph crosses the x-axis with precision, getting the x-intercept value. For our example, the x-intercept is approximately 1.46, which means \( 3^{1.46} \approx 5 \). Understanding how to find this intercept is a valuable skill in solving various types of exponential equations effectively.

Approximation Methods

Approximation methods play an important role when an exact solution is difficult or impossible to find analytically, particularly in the case of exponential equations. In the problem \( 3^x = 5 \), an exact solution is not straightforward; however, an approximation can be achieved using a graphing calculator.

The TRACE function allows you to "trace" along the curve of the graph to pinpoint where it crosses the x-axis. Combined with ZOOM, these tools help achieve an approximation to the desired precision, such as the nearest hundredth. Through approximation, students learn to deal with real-world problems that require estimating values from graphical data. These methods are not only practical but essential for deeper exploration into mathematics.

Graphing Functions

Graphing functions is a foundational skill in mathematics, providing a visual representation of equations and enabling a better understanding of their properties and behavior. For exponential equations like \( 3^x = 5 \), graphing \( f(x) = 3^x - 5 \) offers insight into how the value of the function changes with \( x \).

By graphing a function, you can visually determine key elements like the x-intercept, where the function equals zero. In this activity, graphing helps to approximate the solution by observing where the curve intersects with the x-axis. The graphing process is straightforward with a graphing calculator, allowing students to explore different functions and understand how changes in equations affect their graphs.

• The shape of the graph provides clues about the function's growth or decay.
• Important points, like intercepts, can be easily identified and analyzed.
• Understanding graphing equips students with skills to navigate more complex mathematical concepts.

Graphing functions thus acts as a bridge between abstract equations and tangible understanding, enhancing problem-solving capabilities.