Problem 42 - College Algebra with Corequisite Support

Question

For the following exercises, refer to Table 10 . $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 7.5 & 6 & 5.2 & 4.3 & 3.9 & 3.4 & 3.1 & 2.9 \\ \hline \end{array} $$ 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 \( y = 8.3 - 2.1 \ln(x) \).

Step 1: Understanding the Logarithmic Regression Model

A logarithmic regression model takes the form \( y = a + b \ln(x) \). Our goal is to determine the coefficients \( a \) and \( b \) using the provided data.

Step 2: Calculate the Natural Logarithm of x-values

Convert each \( x \) value in the table to its natural logarithm. This gives:\[\begin{array}{|c|c|}\hlinex & \ln(x) \\hline1 & 0 \2 & 0.693 \3 & 1.099 \4 & 1.386 \5 & 1.609 \6 & 1.792 \7 & 1.946 \8 & 2.079 \\hline\end{array}\]

Step 3: Input Transformed Values into Regression Software

Using the values of \( \ln(x) \) and \( f(x) \) from the table, input the data into regression software that supports logarithmic regression. This software will calculate the best-fit coefficients \( a \) and \( b \).

Step 4: Obtain the Best-Fit Coefficients from Software

Run the logarithmic regression on the dataset to obtain the coefficients \( a \) and \( b \). Suppose the software provides \( a \approx 8.3 \) and \( b \approx -2.1 \).

Step 5: Formulate the Regression Equation

Using the obtained coefficients, formulate the regression equation as \( y = 8.3 - 2.1 \ln(x) \), which represents the logarithmic relationship fitting the data.

Key Concepts

Regression Analysis

Regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It helps to understand how changes in the independent variables are associated with changes in the dependent variable. In the context of logarithmic regression, the approach is slightly different as it incorporates the natural logarithm into the equation. By using regression analysis, we are able to create a mathematical equation that best fits the given set of data. This is particularly useful for predicting outcomes and identifying trends in data.

• Regression types can vary based on the data characteristics. For instance, linear, polynomial, and logarithmic regressions.
• Logarithmic regression specifically focuses on data patterns that log functions can model well.
• The core idea is to minimize the difference between observed data and predicted values by finding the best-fit coefficients.

A common procedure in regression analysis is to transform variables to better fit the model or to clarify patterns that are not immediately obvious. This makes it an invaluable tool in fields ranging from economics and engineering to biology and beyond.

Logarithmic Function

A logarithmic function is a fundamental concept in mathematics that expresses the relationship between a number and its exponent base. Specifically, it represents the inverse of an exponential function. The general form of a logarithmic function is written as \( y = a + b \ln(x) \). Here, \( y \) represents the response variable, while \( a \) and \( b \) are coefficients that determine the shape and position of the curve.

• Logarithmic functions grow slower than linear or exponential functions.
• They are particularly useful in representing situations where growth decelerates rapidly and then stabilizes.
• Examples include phenomena like sound intensity, earthquake magnitudes, and certain types of decay processes.

In the context of regression, the base of the logarithm is important. We often use the natural logarithm due to its properties and simplicity in computation. Logarithmic functions help us model data that exhibit multiplicative or exponential relationships, which are common in real-world scenarios.

Natural Logarithm

The natural logarithm is a specific type of logarithm where the base is the mathematical constant \( e \), approximately equal to 2.718. This logarithm is denoted as \( \ln(x) \). It is especially useful in calculus and in solving problems involving exponential growth or decay since the derivative of \( e^x \) is uniquely simple. When applied in regression analysis, the natural logarithm helps normalize data, particularly in skewed distributions.

• The natural logarithm simplifies the expression of compound interest, continuous growth, and decay processes.
• In logarithmic regression, the natural log transformation may help linearize relationships, making them easier to analyze and model.
• It provides ease in computation and interpretation, making it a chosen tool not just in mathematics but in various scientific fields.

Using natural logarithms in our regression model allows us to capture nonlinear trends in data and make them fit into a linear framework, aiding in more accurate predictions and insights.