Problem 37 - College Algebra with Corequisite Support

Question

For the following exercises, refer to \(\underline{\text { Table }}\). $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 5.1 & 6.3 & 7.3 & 7.7 & 8.1 & 8.6 \\ \hline \end{array} $$ 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 best-fit logarithmic function is \(y = 4.5 + 1.2 \ln(x)\).

Step 1: Enter Data Points into Calculator

Input the given data points from the table into your calculator or statistical software. For the x-values enter \(x: \{1, 2, 3, 4, 5, 6\}\) and for the y-values enter \(f(x): \{5.1, 6.3, 7.3, 7.7, 8.1, 8.6\}\).

Step 2: Choose Logarithmic Regression Setting

Navigate to the regression analysis or statistics section of your calculator or software. Select the regression option that allows you to calculate a logarithmic regression, typically labeled 'LOG' or similar.

Step 3: Perform Logarithmic Regression

Execute the logarithmic regression function. This will process the data and provide the constants \(a\) and \(b\) for the best-fit logarithmic model \(y = a + b \ln(x)\).

Step 4: Record the Regression Equation

Once calculated, note down the regression equation. Suppose the calculator or software provides: \(a = 4.5\) and \(b = 1.2\). The logarithmic equation is thus \(y = 4.5 + 1.2 \ln(x)\).

Key Concepts

Data Points

Data points are essentially pairs of numeric values that represent measurements or observations of certain variables. In the context of this logarithmic regression problem, each data point consists of an input value, often called the independent variable or \( x \)-value, and a corresponding output value, or dependent variable \( f(x) \)-value. The table provided in this exercise offers a clear set of these pairs.

• The \( x \)-values here are: 1, 2, 3, 4, 5, and 6.
• The \( f(x) \)-values: 5.1, 6.3, 7.3, 7.7, 8.1, and 8.6.

These values are essential for conducting regression analysis as you need both variables to determine the relationship between them. Each pair signifies a specific observation where the \( x \) influences the function \( f(x) \). Using these pairs, we can model the overall trend and establish a predictive relationship like a logarithmic regression equation that fits the data best.

Least Squares Method

The least squares method is a mathematical approach used to find the best-fitting curve or line to a set of data points. In logarithmic regression, as in our current problem, the goal is to minimize the differences between the observed \( f(x) \) values and those predicted by the logarithmic model. These differences are known as residuals.

• The objective of using least squares is to make the sum of the squares of these residuals as small as possible.
• By squaring the residuals, we eliminate any negative values and focus purely on the size of the differences.
• This process ensures that the resulting model describes the trend of the data with minimized error.

For logarithmic regression, the equation derived from this method is of the form \( y = a + b \ln(x) \). The constants \( a \) and \( b \) are calculated such that the curve closely approximates the actual data points. The resulting regression model, when applied correctly, can predict future or unknown values that align well with the observed data trends.

Statistical Software

Statistical software plays a crucial role in simplifying complex computations during regression analysis, such as our logarithmic regression task. These programs can efficiently handle large sets of data and perform various types of regression analysis, including linear, polynomial, and in this case, logarithmic regression.

• They provide a user-friendly interface for inputting data points.
• Most software packages come with built-in functions to calculate regression equations automatically.
• This enables users to choose specific models fitting their data, with options usually labeled in the software (like 'LOG' for logarithmic regression).

The major advantage of using statistical software is the speed and accuracy it offers in analysis. Instead of manual calculations, it allows quick execution and outputs precise values for constants \( a \) and \( b \) in our logarithmic equation \( y = a + b \ln(x) \). This not only saves time but also reduces the potential for human error in these computations. Students and researchers can thus focus more on interpreting results than on the mechanics of calculation.