In exponential growth functions, the base of the exponent must be greater than 1. how would the function change if the base of the exponent were 1? how would the function change if the base of the exponent were between 0 and 1?

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in exponential growth functions, the base of the exponent must be greater than 1. how would the function change if the base of the exponent were 1? how would the function change if the base of the exponent were between 0 and 1?

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LectureNotes said in exponential growth functions, the base of the exponent must be greater than 1. How would the function change if the base of the exponent were 1? How would the function change if the base of the exponent were between 0 and 1?

Answer:

In exponential growth functions, the base of the exponent is a crucial factor that determines how the function behaves. Let’s explore the scenarios where the base of the exponent is 1 and where it is between 0 and 1.

1. Base of the Exponent is 1

If the base of the exponent is 1, the exponential function can be expressed as:

f(x) = a \cdot 1^x

Since any number raised to any power of 1 is always 1, this simplifies to:

f(x) = a \cdot 1 = a

In this case, the function is no longer exponential; it becomes a constant function. The value of the function does not change with respect to ( x ) and remains constant at ( a ). This means that instead of showing exponential growth or decay, the function will produce a horizontal line on a graph.

2. Base of the Exponent is Between 0 and 1

If the base of the exponent is between 0 and 1, say ( b ) where ( 0 < b < 1 ), the exponential function can be expressed as:

f(x) = a \cdot b^x

In this scenario, as ( x ) increases, ( b^x ) decreases because the base ( b ) is a fraction less than 1. This results in exponential decay rather than growth. The function will approach zero as ( x ) tends to infinity. Conversely, as ( x ) decreases (becomes more negative), ( b^x ) increases because raising a fraction to a negative power yields a larger number. This means:

  • For positive ( x ), ( f(x) ) decreases exponentially.
  • For negative ( x ), ( f(x) ) increases exponentially.

Graphical Representation:

  • Base of 1: The graph is a horizontal line at ( y = a ).
  • Base between 0 and 1: The graph shows exponential decay, starting from ( a ) when ( x = 0 ) and approaching zero as ( x ) increases.

Summary:

  • Base of 1: The function becomes a constant function f(x) = a.
  • Base between 0 and 1: The function exhibits exponential decay.

Understanding these behaviors is crucial in various applications such as population dynamics, radioactive decay, and financial models, where the base of the exponential function determines the nature of growth or decay.