Problem 63 - A Graphical Approach to College Algebra

Question

Use a graphing calculator to solve each equation. Give solutions to the nearest hundredth. $$2^{-x}=\log _{10} x$$

Step-by-Step Solution

Answer

The solution is approximately \(x = 0.77\).

Step 1: Understand the Equation

We need to solve the equation \(2^{-x} = \log_{10}x\). This means finding the value of \(x\) where the two expressions are equal.

Step 2: Graph the Functions

Using a graphing calculator, plot the two functions \(y = 2^{-x}\) and \(y = \log_{10} x\) on the same coordinate system. Make sure your calculator is set to calculate logarithms with base 10.

Step 3: Find the Intersection Point

On the graph, visually identify the point where the two curves intersect. This intersection point represents the solution to the equation \(2^{-x} = \log_{10} x\).

Step 4: Read the x-coordinate

Using the graphing calculator's functionality, find the exact coordinates of the intersection point. The x-coordinate at this point will be the solution, rounded to the nearest hundredth.

Step 5: Verify the Solution

Confirm that the x-coordinate obtained satisfies the original equation by substituting it back into the expressions for both \(y = 2^{-x}\) and \(y = \log_{10}x\) to ensure they are approximately equal.

Key Concepts

Exponential and Logarithmic Equations

Exponential and logarithmic equations often involve relationships that can be described using exponential and logarithmic functions. In our example, the equation \(2^{-x} = \log_{10}x\) involves an exponential function \(2^{-x}\) and a logarithmic function \(\log_{10} x\). Exponential functions have the form \(b^{x}\), where \(b\) is the base, and logarithmic functions \(\log_{b} x\) are the inverse operations of exponential functions. This means that an exponential function and its corresponding logarithmic function can "undo" each other under certain circumstances.

To solve problems like this, one must analyze both functions independently and then see where they intersect. Finding shared values (often the solution to the equation) can involve techniques such as graphing or algebraic manipulation. In equations that set an exponential function equal to a logarithmic function, graphing is a powerful tool because it helps visualize where both functions yield the same output values. These intersection points provide the solutions to the equation.

Finding Intersection Points

When solving equations containing functions like \(2^{-x}\) and \(\log_{10}x\), one effective approach is to graph both functions and look for their intersection points. These points show where the functions have the same values. Using a graphing calculator can simplify this process greatly.

To graph these functions using a graphing calculator, follow these steps:

• Enter the function \(y = 2^{-x}\) into the calculator under one function option.
• Enter the function \(y = \log_{10} x\) under a separate option.
• Ensure that the graphing window is set appropriately, so both curves are visible. Adjust the window size if necessary to capture the intersection.
• Use the calculator’s intersect feature to pinpoint exactly where these graphs cross.

Once you find the intersection point, the x-coordinate at this intersection represents the solution to the equation. This graphical intersection signifies the x-value at which both original equations have equal outputs.

Rounding Solutions

Rounding solutions is a crucial step in ensuring that your answer is accurate and usable in practical settings. In equations involving functions like those in our problem, final solutions can sometimes involve irrational numbers or long decimals.

To round your solutions effectively to the nearest hundredth:

• Find the x-value from the intersection point using your calculator. This value might be a long decimal.
• Look at the third decimal place. If it is 5 or more, round the second decimal place up by one. If it's less than 5, leave the second decimal place as is.
• For example, if you find \(x = 2.4387\), you would round it to \(2.44\) because the third decimal place (8) is greater than 5.

This rounded value is what you would report in your answer, ensuring it is both precise and concise, as often needed in mathematical solutions and everyday communication.