Problem 462 - Precalculus

Question

For the following exercises, refer to Table 4.28. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline f(x) & {5.1} & {6.3} & {7.3} & {7.7} & {8.1} & {8.6} \\\ \hline\end{array}$$ 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

Use a calculator's logarithmic regression feature to find \(y = a + b \ln(x)\) for the best fit.

Step 1: Organize the Data

List the values provided in the table: \(x = \{1, 2, 3, 4, 5, 6\}\) and \(f(x) = \{5.1, 6.3, 7.3, 7.7, 8.1, 8.6\}\). Keep in mind that we are fitting a model of the form \(y = a + b \ln(x)\).

Step 2: Calculate Logarithms

Calculate \(\ln(x)\) for each value of \(x\):- \(x=1, \ln(1) = 0\)- \(x=2, \ln(2) \approx 0.693\)- \(x=3, \ln(3) \approx 1.099\)- \(x=4, \ln(4) \approx 1.386\)- \(x=5, \ln(5) \approx 1.609\)- \(x=6, \ln(6) \approx 1.792\)

Step 3: Set Up Regression

Use the calculated \(\ln(x)\) values as the independent variable and \(f(x)\) values as the dependent variable in a logarithmic regression analysis. The form of the equation for regression analysis is \(y = a + b \ln(x)\).

Step 4: Perform Regression Calculation

Use a statistical or graphing calculator with regression features to perform a logarithmic regression analysis. Input the \(\ln(x)\) values for \(x\) and \(f(x)\) values for \(y\), and compute the regression, which will provide the best fit coefficients \(a\) and \(b\).

Step 5: Write the Equation

Based on the regression calculation, write down the equation of the form \(y = a + b \ln(x)\) using the coefficients obtained in the previous step.

Key Concepts

Data Analysis

Data analysis is the process of inspecting and transforming raw data to draw useful conclusions. In this context, we have a set of data points related to specific inputs and outputs. The dataset includes input values, known as \( x \), and corresponding function outputs, \( f(x) \).
It's essential to prepare and understand your data before performing any analysis. Organizing and understanding your dataset will significantly help when finding relationships or trends noticeable within it.
The table in the original exercise provides us with an organized dataset that pairs values of \( x \) with their respective \( f(x) \). This preparation is crucial when considering which type of mathematical model or function might accurately represent the relationship between these two variables.
Using tools like graphs or statistical software can help visualize this data, making it easier to decide which type of regression analysis can be used. This educates us about how the data behaves and what patterns to expect.

Logarithmic Function

A logarithmic function is a mathematical function used to describe the relationship between two variables where one variable changes at a rate proportional to its current value. The logarithmic function can be expressed in terms of natural logarithms, denoted as \( \ln(x) \).
In our exercise, we aim to establish a relationship of the form \( y = a + b \ln(x) \). Here, \( y \) represents the output we want to predict or model, while \( x \) is the independent variable under analysis.
The logarithmic function is beneficial in data analysis when the rate of change decreases over time, indicative of many real-world data relationships. It helps linearize exponential growth or decay, thus simplifying complex data patterns into more manageable forms.

Regression Analysis

Regression analysis is a statistical approach to modeling the relationship between a dependent variable and one or more independent variables. In our context, this involves using a logarithmic regression to best fit the provided data points through the model \( y = a + b \ln(x) \).
Performing regression analysis involves several guided steps:

• Identify the independent (\( x \)) and dependent (\( y \)) variables.
• Transform the data using the logarithmic function if necessary.
• Utilize statistical software or calculators capable of regression analysis to determine the coefficients \( a \) and \( b \).

These coefficients represent the intercept and the rate of change respectively, ultimately defining our best-fit line. This line minimizes the distance between the predicted and actual values, thus providing the best approximation given the nature of our data.
Regression is a powerful tool in data analysis, offering insights into data trends and making future predictions based on historical data patterns.