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Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are typically written in the form \( P = P_{0}b^t \), where:

• \( P \) is the amount at time \( t \).
• \( P_{0} \) is the initial amount.
• \( b \) is the base of the exponential, determining growth if \( b > 1 \) or decay if \( b < 1 \).
• \( t \) is the time variable.

In scenarios where the base is \( e \), known as the natural base (approximately 2.71828), the function simplifies the calculus involved in the growth or decay analysis. This is expressed as \( P = P_{0} e^{kt} \), with \( k \) being the growth (\( k > 0 \)) or decay (\( k < 0 \)) rate.
Converting an exponential function to this form allows for easier differentiation and integration, which are key in analysis within sciences and financial studies.

Base conversion in exponential functions is about transferring the base of the exponent to another base, often "\( e \)", the natural base, for ease of analysis. Let's break down how we convert a base, like 0.9 in our exercise, to the natural base.Steps to Convert Base:

• Recognize the existing base: In the function \( P = 174 (0.9)^t \), the base is 0.9.
• Identify the equivalent expression using the natural base "\( e \)": We express 0.9 as \( e^k \).
• Find \( k \) by solving \( 0.9 = e^k \), leading us to \( k = \ln(0.9) \).

This conversion is crucial because it streamlines complex equations, allowing them to be tackled with calculus. The exponential function can now be written as \( P = 174 e^{kt} \), simplifying continuous growth or decay calculations.

Mathematical modeling involves using mathematical concepts and structures to represent real-world situations. In our case, exponential functions model processes like population growth, radioactive decay, or interest calculations.
These models begin with some assumptions:

• The system is self-contained and does not interact with external forces while modeling.
• Changes, like growth or decay, occur at rates consistent with the model.

Using exponential functions in modeling assists in predicting future states based on current data, making it invaluable in science, finance, and engineering. Once we have converted a base to the natural base, we easily apply calculus to derive the system's rate of change. For instance, population growth modeled with \( e \) helps ecologists forecast for management efforts.
If our modeling is incorrect, we must revisit assumptions or data as a misstep in these can lead to large discrepancies in predictions.