Problem 38 - 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 logarithmic function to find the value of the function when \(x=10\)

Step-by-Step Solution

Answer

The value of the function at \( x = 10 \) is approximately 11.4.

Step 1: Understand The Problem

We need to use a logarithmic function to approximate the value of the function \( f(x) \) given in the table for the value of \( x = 10 \).

Step 2: Determine Relationship Type

Since we're asked to use a logarithmic function, it's likely there is a logarithmic relationship between \( x \) and \( f(x) \) in the data. This means we'd assume \( f(x) = a \, \log(x) + b \).

Step 3: Use Data Points to Find Logarithmic Function

To find a logarithmic function, choose two points from the table to establish a model. E.g., using \((1, 5.1)\) and \((2, 6.3)\):\[ f(x) = a \, \log(x) + b \]. Substitute and solve for \(a\) and \(b\).

Step 4: Solve for Constants

Using \(x = 1, f(x) = 5.1\):\[ 5.1 = a \, \log(1) + b \Rightarrow b = 5.1 \].\Using \(x = 2, f(x) = 6.3\):\[ 6.3 = a \, \log(2) + 5.1 \Rightarrow a = \frac{6.3 - 5.1}{\log(2)} \]. Calculate the value of \(a\).

Step 5: Calculate Logarithmic Function

From the above, we've found \( a \approx 1.2 \) (depending on more precise calculation from \( \log(2)\)). Our function is \( f(x) \approx 1.2 \, \log(x) + 5.1 \).

Step 6: Predict Function Value for x=10

Substitute \( x = 10 \) into our function: \[ f(10) = 1.2 \, \log(10) + 5.1 = 1.2 \times 1 + 5.1 = 6.3 + 5.1 = 11.4 \]. This means the approximated value of the function at \( x = 10 \) is 11.4.

Key Concepts

Approximation Methods with Logarithmic Functions

When dealing with complex data and trying to predict values, approximation methods come in handy, especially when using logarithmic functions. These methods help us estimate the output of a function at points where direct data is unavailable. Using the provided table, our goal is to find the function value at \( x = 10 \).
To start, we identify a pattern or trend in the given values. Logarithmic functions are particularly useful because they can model data with a gradually increasing or decreasing rate. In simpler terms, they help in situations where growth slows down over time rather than remaining constant.
By expressing the relationship as \( f(x) = a \log(x) + b \), we create a model which can be an effective approximation tool. The constants \( a \) and \( b \) are derived using known data points from the table, ensuring that our function fits as closely as possible to the actual data values. This fitting process is essential to predicting unknown values accurately.

Mathematical Modeling Using Logarithms

Mathematical modeling is the backbone of predicting future values in any data-driven scenario. By developing a mathematical model, we understand how variables relate to each other. In our case, we use a logarithmic model: \( f(x) = a \log(x) + b \).
This process starts by identifying the relationship type. For the given problem, the data suggests a logarithmic nature. After determining this, we use specific data points—such as \((1, 5.1)\) and \((2, 6.3)\)—to calculate the constants \( a \) and \( b \).
The ability to transform a set of discrete data points into a continuous function model is the key advantage of mathematical modeling. This makes predictions more straightforward and ensures that various real-life applications can be understood through mathematical equations, allowing for better decision-making.

Function Prediction Through Enriched Models

Function prediction allows us to estimate unknown values by relying on a model based on known data. Once our logarithmic function \( f(x) = a \log(x) + b \) is established, we can predict the function's value at any \( x \) by simply substituting the point into the function.
This method is incredibly useful in fields like finance, physics, and any area dealing with large datasets that exhibit a growth trend that diminishes over time. For \( x = 10 \), our calculation yields \( f(10) = 1.2 \times \log(10) + 5.1 = 11.4 \). This predicted value is obtained without needing to gather extensive new data, saving time and resources.
Ultimately, function prediction through logarithmic functions offers a way to extend data interpretation beyond immediate, accessible values, making it a powerful tool in both theoretical and practical applications.