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Chapter 4: Problem 55

Use a graphing utility to solve each equation for \(x.\) $$10=2^{-x}$$

Short Answer

Expert verified
The solution to the equation \(10 = 2^{-x}\) can be approximated using a graphing utility, where the x-intercepts represent the approximate solutions to the original equation. Given the limitations of graphical representation, this solution should be considered as an approximation.

Step by step solution

01

Rewriting for clarity

Firstly, let's rewrite the equation for clarity. The original equation is \(10 = 2^{-x}\), which can be written as \(2^{-x}-10 =0\). This allows the equation to be visualized more naturally as a graph by placing the equation on the \(y\)-axis.
02

Graphing the equation

Next, let's graph the function \(y = 2^{-x}-10\) using a graphing utility. The aim is to look for the x-coordinate of the point where the graph intersects with the x-axis, since that x-coordinate would represents the solution to our original equation.
03

Finding the solution

By observing the graph, we can identify the X-intercepts. These points are the solution to our equation. Note that values may not be exact given the limitations of the graphic representation, in which case, it's better to use these as approximate solutions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
Exponential functions are a type of mathematical function where the variable is an exponent. This means that the function's variable is located in the exponent of a constant base. For example, in the equation \( y = 2^x \), 2 is the base, and \( x \) is the exponent. When you work with exponential functions, you typically encounter a base that's constant, like 2 or 10.Here are some key aspects of exponential functions:
  • Growth or Decay: Exponential functions can model either growth or decay. If the base is greater than 1, like in \( y = 2^x \), the function represents exponential growth. Conversely, if the base is between 0 and 1, like in \( y = \left(\frac{1}{2}\right)^x \), the function models exponential decay.
  • Rapid Change: The unique property of exponential functions is how they change. Small changes in the exponent lead to significant changes in the value of the function. This is why they are often used in real-world applications like population growth, radioactive decay, and interest calculations.
  • No X-intercept: Even though exponential functions may approach zero, they never actually reach zero, meaning the graph doesn't intersect the x-axis unless modified.
Understanding these properties can help when working with exponential functions in equations and graphically analyzing them.
Graphing Utilities
Graphing utilities are digital tools that assist in visualizing mathematical functions. These can be handheld devices like graphing calculators or software applications on computers. They are beneficial in analyzing functions, especially when solving equations graphically.Here's how graphing utilities can be helpful:
  • Visual Representation: They provide a clear picture of how functions behave over different intervals, showcasing properties like intercepts, growth, and behavior at infinity.
  • Ease of Use: Graphing utilities allow us to input complex equations easily and see results immediately, which is especially useful for equations that are difficult to solve algebraically.
  • Finding Intercepts: One crucial use is finding intercepts. By graphing equations such as \( y = 2^{-x} - 10 \), you can identify where the graph intersects the x- and y-axis, helping locate solutions.
For students, mastering graphing utilities is a practical skill, enabling them to explore mathematical concepts interactively.
Solving Equations Graphically
When we solve equations graphically, we aim to find the points where the graph of the function crosses the axes. This method is particularly useful for complicated functions, where algebraic solutions might be cumbersome.Here's how you can solve an equation graphically:
  • Graph the Equation: Using a graphing utility, plot the function. For example, reframe \( 10 = 2^{-x} \) as \( y = 2^{-x} - 10 \) to graph it.
  • Identify Intercepts: Look for the x-intercepts on the graph. These are the \( x \) values where the function crosses the x-axis. In our example, the solution is where \( y = 0 \), thus solving \( 2^{-x} = 10 \).
  • Approximate Solutions: Keep in mind that graphing provides a visual approximation. It might not be exact, so always consider checking and refining using other methods like algebraic solutions if needed.
By solving equations graphically, you gain more insights into the behavior of functions and the nature of their solutions, making this technique an indispensable part of mathematics education.

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