Problem 59 - College Algebra

Question

In an exponential decay function, the base of the exponent is a value between 0 and \(1 .\) Thus, for some number \(b > 1\) , the exponential decay function can be written as \(f(x)=a \cdot\left(\frac{1}{b}\right)^{x} .\) Use this formula, along with the fact that \(b=e^{n},\) to show that an exponential decay function takes the form \(f(x)=a(e)^{-n x}\) for some positive number \(n .\)

Step-by-Step Solution

Answer

The decay function is rewritten as \(f(x) = a(e)^{-nx}\).

Step 1: Understand the given function

First, note the given function: \( f(x) = a \cdot \left(\frac{1}{b}\right)^{x} \). This represents an exponential decay function where the base of the exponent is between 0 and 1.

Step 2: Substitute for decay base

Since \( b > 1 \), \( \frac{1}{b} \) becomes the decay base, satisfying the condition for decay (base between 0 and 1), i.e., \( \left(\frac{1}{b}\right)^{x} = b^{-x} \).

Step 3: Introduce substitution for b

It's given that \( b = e^{n} \). Therefore, substitute \( e^{n} \) for \( b \) into your equation: \( b^{-x} = (e^{n})^{-x} \).

Step 4: Apply the power rule

Using the power rule \( (e^{n})^{-x} = e^{-nx} \), adjust the expression: \( b^{-x} = e^{-nx} \).

Step 5: Substitute and restructure the decay function

Replace \( b^{-x} = e^{-nx} \) back into the decay function: \( f(x) = a \cdot e^{-nx} \).

Step 6: Verify the equivalence

The exponential decay function \(f(x) = a(e)^{-nx}\) is now in the desired form, showing that it features an exponential decay with decay base \(e\) and exponent \(-nx\).

Key Concepts

Exponential Function

An exponential function is a mathematical expression in the form of \( f(x) = a \cdot b^x \) where \( a \) is a constant, \( b \) is the base, and \( x \) is the exponent or power. This type of function shows either exponential growth or decay depending on the value of \( b \). - **Exponential Growth**: This occurs when the base \( b > 1 \). As \( x \) increases, the value of \( f(x) \) grows rapidly.- **Exponential Decay**: This takes place when the base \( 0 < b < 1 \). As \( x \) increases, \( f(x) \) decreases swiftly.Understanding the nature of the base is crucial in determining the behavior of the exponential function. Exponential functions are widely used in various fields like finance, science, and engineering to model real-world phenomena.

Power Rule

The power rule is a critical tool in calculus for differentiating functions of the form \( x^n \), where \( n \) is any real number. However, in the context of exponential functions, it involves manipulating powers of bases:- The **power rule** states that for any base \( b \), \( (b^m)^n = b^{m \cdot n} \). This means when you have a power raised to another power, you multiply the exponents together.In exponential decay functions, applying the power rule is essential for simplifying expressions and transforming bases, especially when substituting elements like \( b = e^n \) into the function. It helps reconfigure the function while maintaining its integrity, shifting it into a more comprehendible form, such as \( e^{-nx} \).

Base of Exponent

The base of an exponent in an exponential function determines the function's overall behavior, whether it rises or falls. When discussing exponential decay, the base has unique characteristics:- **Base Between 0 and 1**: For decay, the base must satisfy \( 0 < b < 1 \), indicating a decreasing function.- **Substituting Base with \( e \)**: Sometimes, expressions are comfier to work with when rewritten using \( e \), Euler's number (~2.718), due to its unique mathematical properties: for large, complex equations as seen in calculus.Rewriting the decay base \( \left(\frac{1}{b}\right)^x \) takes advantage of the identity \( b = e^n \), aligning with the natural base, \( e \), which plays a pivotal role in simplifying calculations and analysis of exponential decay functions naturally and elegantly.

Substitution Method

The substitution method is a mathematical technique for simplifying equations and expressions through introducing alternative components or variables. In exponential decay functions:- **Introducing \( e^n\)**: If given \( b = e^n \), substituting \( e^n \) for \( b \) in the decay function aids in simplifying and restructuring the expression.- **Reduced Complexity**: This method reduces complexity by aligning terms into recognizable forms like \( e^{-nx} \), cutting down cumbersome steps during manipulation.This method is not only helpful in restructuring equations but also facilitates deeper understanding by allowing us to see connections between different mathematical models and approaches, reinforcing the grasp on decay behavior and exponential phenomena.