Problem 25 - Applied Calculus

Question

In Problems \(25-28,\) put the functions in the form \(P=P_{0} e^{k t}\). $$P=15(1.5)^{t}$$

Step-by-Step Solution

Answer

The function can be written as \(P = 15 e^{t \ln(1.5)}\).

Step 1: Understand the Given Function

We are given a function in the form \(P = 15(1.5)^{t}\). Our task is to express this function in the form \(P = P_0 e^{kt}\). Here, \(P_0\) is the initial value of the function and \(e^{kt}\) represents the exponential growth or decay, with \(k\) as the growth/decay rate.

Step 2: Identify the Initial Value \(P_0\)

The initial value \(P_0\) is the coefficient in front of the expression involving \(t\). In the given function \(P = 15(1.5)^{t}\), we identify \(P_0 = 15\).

Step 3: Convert Base to Natural Exponential

To convert the base of \(1.5\) to a base \(e\), we use the natural logarithm: \(1.5 = e^{\ln(1.5)}\). So we can rewrite \((1.5)^{t}\) as \((e^{\ln(1.5)})^{t} = e^{t \ln(1.5)}\).

Step 4: Rewrite the Function in the Desired Form

Substitute the expression from Step 3 back into the function: \(P = 15 \cdot e^{t \ln(1.5)}\). This is now in the form \(P = P_0 e^{kt}\) with \(P_0 = 15\) and \(k = \ln(1.5)\).

Key Concepts

Natural Logarithm

In the world of mathematics, the natural logarithm is a fundamental concept that simplifies the process of dealing with exponential functions. The natural logarithm, often represented as \( \ln(x) \), is the power to which the base \( e \) (approximately equal to 2.718) must be raised to produce the number \( x \). It is the inverse operation of exponentiation with a base \( e \). This concept is crucial when we need to convert expressions with arbitrary bases to those involving \( e \).
In this exercise, we have the base of 1.5 and need to convert it into base \( e \). We do this through the natural logarithm, expressing 1.5 as \( e^{\ln(1.5)} \). This approach is not just cosmetic: it transforms the expression into one that reflects the continuous growth processes, such as compound interest or population growth. It provides a way to simplify and standardize exponential equations, making them easier to manipulate and analyze.

Exponential Growth

Exponential growth is a critical concept when dealing with processes that increase steadily at a consistent rate over time. In this context, growth is not linear but rather accelerates as time passes, making small initial increases balloon into much larger increments.
We can observe this in functions structured as \( P = P_0 e^{kt} \), where \( P \) represents the quantity of interest that grows exponentially from the initial value \( P_0 \). The factor \( e^{kt} \) accounts for growth over time, where \( k \) is a constant representing the growth rate.
An example of exponential growth is the compounding interest on an investment, where the amount of interest earned increases over time so long as the interest remains part of the principal.
In the exercise, transforming the base from 1.5 to \( e \) allows us to capture the exponential nature of growth that is more universally applicable due to the natural exponential base \( e \).

Growth Rate

The growth rate in exponential functions is a key parameter that dictates how fast the quantity of interest increases over time. Represented by the symbol \( k \) in the expression \( P = P_0 e^{kt} \), it is a multiplier on the time variable \( t \).
A positive \( k \) value results in exponential growth, while a negative \( k \) represents decay. The growth rate \( k \) essentially communicates the percentage change in \( P \) over one time unit.
In our example function, \( P = 15(1.5)^t \), converting it to \( P = 15e^{t \ln(1.5)} \) provides \( k \) as \( \ln(1.5) \), highlighting the transformation from a base of 1.5 to approximately 0.405 using the natural logarithm.

• For growth: Higher \( k \) values mean faster growth.
• The unit of \( k \) is typically in terms of inverse time, aligning with how frequently the growth is compounded.

Understanding \( k \) helps us predict the future behavior of the system described by the exponential function.

Initial Value

In exponential functions, the initial value \( P_0 \) is the starting point of the quantity of interest before any growth occurs. It provides a reference point from which exponential changes are calculated, and it remains constant regardless of the time elapsed.
In the given function \( P = 15(1.5)^t \), it is straightforward to see that \( P_0 = 15 \). This initial value sets the baseline for the growth tracked by the exponential term \( e^{kt} \).
Understanding the initial value is essential as it determines the magnitude of growth or decay reflected by the exponential component. If the initial value were higher or lower, it would proportionally affect the resulting behavior of the function. The initial value is crucial for real-world applications like:

• Population studies, where \( P_0 \) might be the population size at the start of a given period.
• Financial forecasts, where it could represent an account's principal before interest accumulation.

Knowing \( P_0 \) allows us to grasp the scale of the system under observation and anticipate its evolution over time.