Problem 38 - College Algebra

Question

For the following exercises, refer to Table 9. $$\begin{array}{ccccccc}{x} & {1} & {2} & {3} & {4} & {5} & {6} \\ {f(x)} & {5.1} & {6.3} & {7.3} & {7.7} & {8.1} & {8.6}\end{array}$$ Use the logarithmic function to find the value of the function when \(x=10\) .

Step-by-Step Solution

Answer

Using a logarithmic model, \(f(10)\) is approximately 8.05.

Step 1: Identify the Function Type

The problem mentions using the logarithmic function. The values of \(f(x)\) increase slowly, suggesting a possible logarithmic relationship with \(x\). We will fit a logarithmic model to predict the value of \(f(x)\) for \(x=10\).

Step 2: Fit the Logarithmic Model

A logarithmic model is typically of the form \(f(x) = a + b \ln(x)\), where \(a\) and \(b\) are constants. Use the given data points to determine \(a\) and \(b\) using regression methods or by manually solving it using two known points for simplicity.

Step 3: Solve for Constants

Select two pairs of data, such as \((x_1, f(x_1)) = (1, 5.1)\) and \((x_3, f(x_3)) = (3, 7.3)\). Setup the equations \(5.1 = a + b \ln(1)\) and \(7.3 = a + b \ln(3)\). Solve these simultaneous equations:1. \(5.1 = a\) since \(\ln(1) = 0\).2. Substitute \(a\) in the second equation: \(7.3 = 5.1 + b \ln(3)\).3. Solve for \(b\): \(b = \frac{7.3 - 5.1}{\ln(3)} \approx 1.2769\). Thus, \(a = 5.1\), \(b \approx 1.2769\).

Step 4: Substitute into the Model

Substitute \(a = 5.1\) and \(b \approx 1.2769\) into the model equation to get \(f(x) = 5.1 + 1.2769 \ln(x)\).

Step 5: Calculate \(f(10)\)

Now calculate \(f(10)\) using the model: \[ f(10) = 5.1 + 1.2769 \ln(10) \approx 5.1 + 1.2769 \times 2.3026 \approx 8.05 \]

Step 6: Interpret the Result

The calculated value \(f(10) \approx 8.05\) suggests that at \(x = 10\), the function \(f(x)\) is approximately 8.05 based on the logarithmic model derived from the data.

Key Concepts

Regression Methods

When analyzing data, regression methods are vital tools that help us identify relationships between variables. Among various techniques, logistic regression and polynomial regression are popular.
In the context of this exercise, we used a logarithmic regression method. This process involves fitting a logarithmic function to a set of data points to represent a plausible relationship between variables.
Here’s how it works:

• First, hypothesize a mathematical function, usually based on the pattern of data growth or decay. For our exercise, that is a logarithmic form: \(f(x) = a + b \ln(x)\).
• Next, determine the constants \(a\) and \(b\) by using known data points. This can be done via solving equations derived from a pair of different data points.
• Finally, use these constants to predict unknown values. In our problem, after calculating \(a\) and \(b\), the values were used to predict \(f(10)\).

In practice, regression methods allow for more precise predictions and can uncover relationships that are not immediately obvious.

Data Analysis

Data analysis is the process of inspecting, cleaning, transforming, and modeling data. By doing so, we aim to discover useful information, draw conclusions, and support decision-making.
For this particular exercise, data analysis involved observing the provided table and using appropriate concepts to understand the trend. Here's how you can do it:

• Start by plotting the data points if possible. Visual representations often reveal underlying patterns or suggest suitable models.
• Determine the type of relationship between the data points. Is it linear, exponential, logarithmic, or something else?
• In this scenario, the data increment suggests a logarithmic relationship, thus leading to the use of a logarithmic regression model.
• Data analysis also involves interpreting results, ensuring that the chosen model fits well, and making predictions based on this model.

By thoroughly conducting data analysis, you not only solve the current problem but also enhance your skills in identifying patterns, leading to more robust predictions in different contexts.

Mathematical Modeling

Mathematical modeling is about creating abstract representations of real-world phenomena using mathematical concepts and language. In our exercise, we create a model to predict values of a function.
Here's how mathematical modeling is used:

• Identify the characteristics of your data and select the most suitable model. In this case, a logarithmic model seemed sensible given the nature of the dataset.
• Mathematical models like \(f(x) = a + b\ln(x)\) offer a formulaic representation. They help to approximate real values and make predictions.
• Substitute known values to solve for any constants needed for your specific model. In our exercise, this involved calculating \(a = 5.1\) and \(b \approx 1.2769\).
• Utilize the model to determine unknown values (e.g., \(f(10)\) in this exercise), thereby gaining insights that align with the initial data trend.

Remember, effective mathematical modeling involves constant testing and refining. This ensures the model remains relevant and accurately reflects the behaviors or patterns it represents.