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Exponential growth occurs when the base of the exponential function, denoted as \(b\), is greater than 1. In this case, the value of the function increases rapidly as the exponent grows. This sort of growth is often observed in populations, investments, and certain natural phenomena.
For instance, in the function \(N=50 \cdot 2.5^{T}\), the base \(b\) is 2.5. Since 2.5 is greater than 1, no matter what value \(T\) takes (as long as \(T > 0\)), the function will be growing exponentially.
Common characteristics of exponential growth include:

• Rapid increase over time
• The initial quantity is multiplied by a consistent factor over equal time periods
• Doubling times where the quantity doubles after a specific period

Understanding this growth pattern is essential in fields such as finance, biology, and physics, where predicting the future value of quantities based on their growth rates is crucial.

Exponential decay happens when the base of the exponential function is between 0 and 1. This indicates a gradual decrease over time.
For example, consider the function \(g(z)=\left(3 \cdot 10^{5}\right) \cdot(0.8)^{z}\). Here, the base \(b\) is 0.8. Since 0.8 is less than 1 but greater than 0, the function's value decreases as \(z\) increases.
Key features of exponential decay include:

• Steady decrease over time
• The initial quantity decays by a consistent factor in equal time intervals
• Half-lives where the quantity is halved after a specific period

Exponential decay is typically seen in areas like radioactive decay, cooling of objects, and depreciation of assets. Understanding it helps predict how quickly something decreases over time.

Analyzing exponential functions involves determining whether the function represents growth or decay.
First, identify the base \(b\). If \(b > 1\), the function shows exponential growth. If \(0 < b < 1\), it shows exponential decay.
Let's illustrate with the following examples from the exercise:
- In \(y=264 \left(\frac{5}{2}\right)^{x}\), \(b= \frac{5}{2} = 2.5\). Since 2.5 > 1, the function represents exponential growth.
- For \(f(T)=\left(1.5 \cdot 10^{11}\right) \cdot (0.35)^{T}\), \(b=0.35\). Since 0.35 is between 0 and 1, it represents exponential decay.
To analyze these functions effectively, always:

• Determine the base \(b\)
• Check if \(b > 1\) for growth or \(0 < b < 1\) for decay
• Consider real-world contexts to understand implications better

This systematic approach helps clarify whether the value of the function will increase or decrease over time.