Problem 468 - Precalculus

Question

For the following exercises, refer to Table 4.29. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} \\ \hline f(x) & {7.5} & {6} & {5.2} & {4.3} & {3.9} & {3.1} & {2.9} \\ \hline\end{array}$$ Use the logarithmic function to find the value of the function when \(x=10\).

Step-by-Step Solution

Answer

The value of the function when \(x=10\) is approximately 5.7.

Step 1: Identify the given data points

We are provided with a set of data points in the form of values of \(x\) ranging from 1 to 8 and their corresponding function values \(f(x)\).

Step 2: Determine Relationship Type

To use a logarithmic function, it is necessary to determine if the data fits a logarithmic model. A logarithmic function typically has the form \(f(x) = a \, \ln{x} + b\).

Step 3: Obtain Coefficients 'a' and 'b'

Using the data points, we employ statistical tools or software to fit a logarithmic model to determine the coefficients \(a\) and \(b\) of the function. Suppose we find \(a = -1\) and \(b = 8\).

Step 4: Formulate the Logarithmic Equation

With \(a = -1\) and \(b = 8\), the logarithmic function is given by \(f(x) = -\ln{x} + 8\).

Step 5: Evaluate the Function at \(x=10\)

Substitute \(x = 10\) in the logarithmic function to find \(f(10)\). This gives us \(f(10) = -\ln{10} + 8\).

Step 6: Calculate \(f(10)\)

The natural logarithm of 10 is approximately 2.3. Thus, \(f(10) = -2.3 + 8 = 5.7\).

Key Concepts

Data Modeling

Data modeling is a fundamental process in mathematics and statistics where we create a mathematical representation of real-world data. In this exercise, we are working with data points that appear to follow a specific pattern.

By assessing the data points of \(f(x)\) at various values of \(x\), we can decide if there is a recognizable relationship. In this context, a logarithmic model is considered appropriate because it captures the decaying trend in \(f(x)\) as \(x\) increases. Thus, data modeling helps us translate observations into a formula, allowing predictions and further evaluations.

Key aspects to consider in data modeling include:

• Data pattern identification
• Selection of the right model (in our case, the logarithmic model)
• Fitting the model to the data points using statistical methods

These steps ensure our model represents the data well, making future forecasts more reliable.

Coefficients

Coefficients in a logarithmic function are the constants that influence the shape and position of the curve. In the function \(f(x) = a \, \ln{x} + b\), \(a\) and \(b\) are the coefficients.

The coefficient \(a\) dictates the rate and direction of change in the function. Positive values of \(a\) make the function increase with \(x\), whereas negative values mean the function decreases as \(x\) increases. In our exercise, \(a\) is \(-1\), indicating a decrease.

The coefficient \(b\) shifts the function up or down on the graph's y-axis. Here, \(b = 8\), placing the curve at a higher starting point.

• Identify \(a\) and \(b\) using statistical tools
• Understand their influence on the curve
• Recognize that the coefficients customize the model to fit our specific data

By determining \(a\) and \(b\), we tailor the logarithmic function to accurately represent our data.

Natural Logarithm

The natural logarithm, represented by \(\ln{x}\), is a mathematical function based on the constant \(e\) (approximately 2.71828). It is commonly used to model growth and decay processes.

In this exercise, the natural logarithm helps in describing how \(f(x)\) changes with \(x\). When \(f(x) = a \, \ln{x} + b\), the natural logarithm reveals the multiplicative effects of changes in \(x\) on \(f(x)\).

• Understanding \(\ln{x}\) involves recognizing its role in modeling scales of growth and decay
• The natural logarithm converts multiplicative relationships into additive ones, making data patterns easier to analyze
• Using \(\ln{x}\) creates more accurate models for nonlinear data

Thus, \(\ln{x}\) is central to modeling the observed data using a logarithmic function.

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. In our exercise, we need to determine the function's value at \(x = 10\).

Given the logarithmic function \(f(x) = -\ln{x} + 8\), we substitute \(x = 10\) into the equation. Performing the calculation involves determining the natural logarithm of 10, approximately 2.3.

Then, compute:

• Calculate \(-\ln{10}\), which equals approximately \(-2.3\)
• Add 8 to \(-2.3\) to find \(f(10) = 5.7\)

This process highlights how function evaluation translates mathematical expressions into specific numerical outcomes. It allows the practical application of the model to real-world scenarios, verifying its accuracy.