Problem 12 - College Algebra with Corequisite Support

Question

Rewrite the exponential model \(A(t)=1550(1.085)^{x}\) as an equivalent model with base \(e\). Express the exponent to four significant digits.

Step-by-Step Solution

Answer

The model with base \( e \) is \( A(t) = 1550 e^{0.08107x} \).

Step 1: Understand the Exponential Model

The given model is \( A(t) = 1550(1.085)^{x} \). Here, \( 1550 \) is the initial amount, and \( 1.085 \) is the growth factor that changes with respect to \( x \). We need to rewrite this model using the natural base \( e \).

Step 2: Express Using Natural Base

To express an exponential function in terms of \( e \), we use the property \( a^x = e^{x \ln a} \). Here, \( a = 1.085 \). Therefore, we need to express \( (1.085)^x \) as \( e^{x \ln(1.085)} \).

Step 3: Calculate the Exponent

Find \( \ln(1.085) \) using a calculator. The natural logarithm of 1.085 is approximately 0.081067. Thus, the equivalent expression \( (1.085)^x \) can be rewritten as \( e^{0.081067x} \).

Step 4: Write the Equivalent Exponential Model

Rewriting the entire model, we get the equivalent in base \( e \) as \( A(t) = 1550 e^{0.081067x} \). The exponent is expressed to four significant digits.

Key Concepts

Exponential Growth

Exponential growth is a fascinating mathematical concept that shows how quantities increase over time. It is often seen in populations, investments, and particular scientific processes. In an exponential growth model, the amount grows at a constant percentage rate over equal time intervals. The formula usually looks like this:

• \( A(t) = A_0 b^t \)

Here, \( A(t) \) represents the amount at time \( t \), \( A_0 \) is the initial amount, and \( b \) is the growth factor. A growth factor larger than 1 indicates an increase over time. This constant rate of growth leads to rapid increases as the time intervals accumulate.

Natural Base e

The natural base, denoted as \( e \), is approximately equal to 2.71828 and is base of the natural logarithm. It is a transcendental number, much like \( \pi \), which means it cannot be expressed as a simple fraction. \( e \) is an important constant in mathematics, especially in calculus and complex numbers. Using \( e \) in models allows for natural exponential functions, which are pivotal in describing continuous growth processes.

• A natural exponential function has the form \( e^{x} \).
• This form is often preferred for continuous growth models because it allows easier differentiation and integration.

In our exercise, we rewrote the base from 1.085 to \( e \), which helped in understanding the growth in terms of natural logarithms.

Logarithms

Logarithms are the mathematical inverse of exponential functions, serving to "undo" exponential growth and decay calculations. When we encounter expressions like \( a^x = n \), logarithms help us find \( x \). For example, the logarithm with base \( 10 \) is usually denoted as \( \log(x) \), while the natural logarithm is given by \( \ln(x) \), which means log base \( e \). In the process of converting our growth function \((1.085)^x \) to base \( e \), we used the natural log, noting:

• \( a^x = e^{x \ln a} \)

This conversion uses the property of logarithms to reframe the exponential expression in terms of \( e \), thereby simplifying mathematical operations on such expressions.

Mathematical Models

Mathematical models are simplified representations of real-world processes using mathematical language and expressions. They help us understand complex systems by focusing on essential behaviors and relationships. Models such as the one in our exercise, \( A(t) = 1550(1.085)^{x} \), are typically:

• Used to predict future conditions and scenarios.
• Flexible to accommodate different initial amounts or growth rates.

Rewriting such models using the natural base \( e \) allows for more straightforward analysis, particularly when dealing with growth or decay problems in continuous time. It can also help make predictions more intuitive and harmonized with continuous mathematical frameworks like calculus.