Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, refer to \(\underline{\text { Table }} 7 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 1125 & 1495 & 2310 & 3294 & 4650 & 6361 \\ \hline \end{array} $$ Write the exponential function as an exponential equation with base \(e\).

Step-by-Step Solution

Answer

Perform linear regression on \((x, \ln(f(x)))\) to find \(a\) and \(b\) for \(f(x) = a \cdot e^{bx}\).

Step 1: Understanding Exponential Functions

An exponential function with base \(e\) is typically expressed as \(f(x) = a \cdot e^{bx}\), where \(a\) and \(b\) are constants. To determine \(a\) and \(b\), you need to examine the function values at different points \(x\).

Step 2: Convert to Linear Form

For exponential functions \(f(x) = a \cdot e^{bx}\), taking the natural logarithm of both sides gives \(\ln(f(x)) = \ln(a) + bx\). This is the equation of a line where \(\ln(a)\) is the y-intercept and \(b\) is the slope.

Step 3: Use Table Data

Rewrite the function values \(f(x)\) from the table to their natural logs: \(\ln(1125), \ln(1495), \ln(2310), \ln(3294), \ln(4650), \ln(6361)\). This will give a linear dataset.

Step 4: Perform Linear Regression

Apply linear regression to the transformed data \((x, \ln(f(x)))\) from step 3. This will help determine the best-fit line of the form \(\ln(f(x)) = \ln(a) + bx\), which provides \(\ln(a)\) and \(b\).

Step 5: Determine \(a\) and \(b\)

Using the linear regression results, solve for \(a\) by taking the exponential of the y-intercept: \(a = e^{\ln(a)}\), and use the slope for \(b\).

Step 6: Write the Exponential Equation

Substitute \(a\) and \(b\) back into the exponential function \(f(x) = a \cdot e^{bx}\) using the values obtained from step 5 to form the complete equation.

Key Concepts

Exponential Equation

Understanding exponential equations is key for handling growth models in math and real-life applications. An exponential equation represents a situation where a quantity grows or decays at a constant rate. These equations typically take the form \(f(x) = a \cdot b^x\). In the context of base \(e\), a natural exponential function takes the form \(f(x) = a \cdot e^{bx}\). Here, \(a\) represents the initial amount or starting value, and \(b\) represents the rate of growth or decay.

To solve problems using exponential equations, you need to find these constants, \(a\) and \(b\). This involves analyzing data to see how values change with respect to \(x\). Exponential equations are powerful for modeling scenarios where change compounds over time, such as interest calculations, population growth, or radioactive decay.

Base e

The mathematical constant \(e\), approximately equal to 2.718, is fundamental in nature. Known as Euler's number, it's the base of natural logarithms. This constant frequently appears in calculus, particularly in growth and decay processes.

In exponential functions with base \(e\), such as \(f(x) = a \cdot e^{bx}\), the use of \(e\) makes calculations involving continuous growth or decay straightforward. The nice property of \(e\) is its natural scaling behavior. This makes it ideal for describing systems where change is both proportional to the current amount and continuous. Learning to work with \(e\) lets you better understand seamless transitions, like those in financial calculations and scientific models.

Overall, base \(e\) expedites mathematical modeling by representing continuous processes more naturally.

Linear Regression

Linear regression is a statistical method used to find the relationship between a dependent variable and one or more independent variables. It's particularly useful when transforming exponential data using logarithms. After converting your exponential equation form using the natural logarithm, it becomes linear.

In the context of our task, we apply linear regression to the transformed data set, \((x, \ln(f(x)))\). By plotting these pairs, you might notice a linear trend. Linear regression finds the best-fit line that models this linear relationship, allowing you to determine \( \ln(a) \) and \(b\).

This step helps us solve for the parameters of the original exponential function, as it shows how changes in \(x\) affect \(\ln(f(x))\). Mastering linear regression opens up capabilities to model, analyze, and foresee various trends in data science and statistics.

Natural Logarithm

The natural logarithm, denoted \(\ln\), is the logarithm with the base \(e\). It transforms complex multiplicative relationships into simpler additive ones. In mathematics, \(\ln\) is particularly powerful for solving exponential equations as it linearizes them.

When dealing with exponential functions like \(f(x) = a \cdot e^{bx}\), the natural logarithm is used to convert the exponential form to a linear form. This is achieved through the identity \(\ln(f(x)) = \ln(a) + bx\). This transformation allows you to apply linear regression techniques, making it easier to ascertain the best-fit line and determine key parameters.

The natural logarithm simplifies analysis by providing a constant rate of change. It is invaluable in calculus, solving equations involving \(e\), and understanding phenomena in the natural sciences and finance.