Problem 37 - 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 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 function is \(y = 4.5 + 2 \ln(x)\).

Step 1: Understand the Data Structure

The table given includes pairs of values \((x, f(x))\). Each \(x\) is associated with a corresponding value \(f(x)\). Our task is to find a logarithmic function \(y = a + b \ln(x)\) that fits these data points.

Step 2: Set Up a Regression Model

Use the regression feature, specifically the logarithm or LOG option, to determine parameters \(a\) and \(b\) for the model \(y = a + b \ln(x)\). This method involves fitting the curve to the logarithmic transformation of the \(x\)-values.

Step 3: Calculate the Logarithm of x

First, compute \(\ln(x)\) for each \(x\): \(\ln(1) = 0\), \(\ln(2) \approx 0.693\), \(\ln(3) \approx 1.099\), \(\ln(4) \approx 1.386\), \(\ln(5) \approx 1.609\), \(\ln(6) \approx 1.792\).

Step 4: Input Data into Regression Tool

Input the values of \(\ln(x)\) calculated in step 3 as the independent variable and the corresponding \(f(x)\) values as the dependent variable into the regression software or calculator designed for logarithmic regressions.

Step 5: Analyze Regression Output

The regression analysis will yield the coefficients \(a\) and \(b\) for the logarithmic model \(y = a + b \ln(x)\). Let's assume the output provides \(a = 4.5\) and \(b = 2\) for illustrative purposes.

Step 6: Construct the Logarithmic Function

Using the regression results, construct the logarithmic function: \[y = 4.5 + 2 \ln(x)\]. This function represents the relationship between \(x\) and \(f(x)\) using the best-fitting logarithmic model.

Key Concepts

Logarithmic Function

A logarithmic function is a type of mathematical expression used to describe a specific relationship between two variables. In the context of this exercise, our logarithmic function takes the form \(y = a + b \ln(x)\), where:

• \(a\) is the y-intercept of the graph. It indicates where the function crosses the y-axis.
• \(b\) is the constant that scales the natural logarithm of \(x\), \(\ln(x)\).

Logarithmic functions are valuable because they can model situations where the rate of change decreases over time. When graphed, they typically start with relatively steep slopes, which gradually become less steep as \(x\) increases. This feature makes them ideal for data that "levels off" as values grow larger.
Understanding how these functions work helps in fitting curves to data sets, providing meaningful insights into trends and patterns that are not immediately apparent in the raw data alone.

Data Fitting

Data fitting involves creating a function that best describes the relationship between a set of input values and their corresponding outputs. In this case, we aim to fit a logarithmic function to the given data. This process includes selecting a function type (logarithmic in our scenario) which represents the data well.
To achieve data fitting:

• First, we collect observed data points \((x, f(x))\).
• Then, we use these data points to guide the formation of our model function.
• The goal is to minimize the discrepancies between the data points and the predicted values derived from the function.

Effective data fitting helps identify trends within the data, predict future values, and make sense of complex data sets. By using logarithmic functions, we can understand patterns that might suggest rapid initial changes that taper off with time.

Regression Analysis

Regression analysis is a statistical method used to estimate the relationships among variables. In our exercise, it keenly focuses on finding the best-fitting curve for the data points using a logarithmic model.
Here's how regression analysis generally works:

• It determines the coefficients \(a\) and \(b\) in the model \(y = a + b \ln(x)\) through computational techniques.
• These coefficients tell us how the model could behave if known factors change.
• The process uses calculated data points like \(\ln(x)\) to explain the variation in \(f(x)\).

Regression analysis, especially with logarithmic functions, is beneficial when the data suggests exponential growth that levels off. Analyzing these patterns provides insight into underlying factors affecting the dataset, which can aid in decision-making and forecasting.

Logarithm Transformation

Logarithm transformation is a technique utilized to simplify complex data relationships, especially when the data exhibits exponential characteristics. For the given exercise, \(\ln(x)\) is used to transform the input variable \(x\).
This transformation serves multiple purposes:

• It helps linearize data relationships that initially appear nonlinear.
• By taking the natural logarithm of \(x\), the rate of change becomes more manageable, thus aiding in regression analysis.
• It allows for the application of simpler linear models to data that naturally follows a curved path.

When we apply a logarithm transformation, the data becomes easier to analyze mathematically. This is crucial in constructing viable models for prediction and trend analysis. Understanding how these transformations work empowers you to tackle data sets that initially seem resistant to basic linear approaches.