Problem 60 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The solution is approximately 0.53.

Step 1: Understand the Equation

The given equation is \(2^{-x} = \log x\). Our goal is to find the value of \(x\) where these two expressions are equal. We'll use a graphing calculator to plot both functions individually.

Step 2: Plot the Functions

Using a graphing calculator, enter the functions \(y_1 = 2^{-x}\) and \(y_2 = \log x\). Make sure your calculator is set to the correct mode for the logarithm base, which is base 10 for \(\log x\).

Step 3: Find the Intersection Points

On the graphing calculator, use the intersection function to find where the graphs of \(2^{-x}\) and \(\log x\) intersect. This will give the solution to the equation. The intersection points show the values of \(x\) where \(2^{-x} = \log x\).

Step 4: Record the Solutions

The calculator finds the intersection points accurately. Record the \(x\)-values of each intersection point. These are the solutions to the equation, which should be rounded to the nearest hundredth.

Key Concepts

Exponential Functions

Exponential functions have the form \(a^x\), where \(a\) is a constant and \(x\) is the variable. They are characterized by a rapid increase or decrease in values. For example, the function \(2^{-x}\) is an exponential function where the base is 2, and the exponent is the negative variable \(-x\). This means as \(x\) increases, the value of \(2^{-x}\) decreases, creating a graph that falls towards zero.
The properties of exponential functions include:

• Constant Ratio: The value changes at a constant multiplicative rate.
• Nonzero y-intercept: When \(x = 0\), the function evaluates to the base raised to the zero exponent, which is always 1.
• One-to-one Function: Each input \(x\) produces a unique output \(a^x\).

Because of these features, exponential functions like \(2^{-x}\) are useful in modeling many real-world phenomena, such as radioactive decay and population declines.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The function \(\log x\) is the logarithm of \(x\) with base 10, also known as the common logarithm. Logarithmic functions grow very slowly compared to exponential functions, tending to infinity as \(x\) increases. This means that values of \(x\) greater than 1 will give positive logarithmic outputs, while values between 0 and 1 produce negative outputs.
Key aspects of logarithmic functions include:

• Inverse Relationship: If \(y = a^x\), then \(x = \log_a(y)\). In our equation, \(\log x\) is trying to find the exponent to which 10 should be raised to get \(x\).
• Domain: Defined only for positive \(x\) values because you cannot take the logarithms of zero or negative numbers.
• Slow Growth: This makes them useful for suppressing large number scales.

By understanding the behavior of \(\log x\), you can predict trends over time in situations like measuring the intensity of earthquakes or the acidity levels in chemistry.

Equation Solving

Equation solving involves finding the values where two mathematical expressions are equal. In the case of \(2^{-x} = \log x\), solving such equations often requires graphical methods, especially when exact algebraic solutions are complex or impossible.
To solve this equation:

• Graphically Approach: Plot each side of the equation as separate functions, \(y_1 = 2^{-x}\) and \(y_2 = \log x\), using a graphing calculator.
• Intersection Points: Utilize tools to highlight where the curves intersect. The x-coordinate of these points provides the solution to the equation \(2^{-x} = \log x\).
• Accuracy Practice: To ensure precision, record values to the nearest hundredth, ensuring a practical balance between accuracy and usability.

These intersections represent solutions where both expressions are equal, thus providing the desired solutions to the equation.