The base of an exponential function represents the factor by which the function's value changes for each unit change in the input variable. In simpler terms, it's the foundation upon which the exponential growth or decay is built.
Understanding the Base
The base, typically denoted as b in the general form of an exponential function f(x) = ab^x, plays a critical role in determining the function's behavior. According to the reference information, the base b* must be a number greater than 0, except for 1. Let's break down what this means:
- b > 1: If the base is greater than 1, the function models exponential growth. This means the function's value increases rapidly as x increases.
- 0 < b < 1: If the base is between 0 and 1, the function models exponential decay. This means the function's value decreases rapidly as x increases, approaching zero.
The base b, serves as a starting point for calculating the growth or decline. It is the constant multiplier.
Examples
| Function | Base (b) | Behavior | Explanation |
|---|---|---|---|
| f(x) = 2^x | 2 | Exponential Growth | For every increase of 1 in x, the function's value doubles. |
| f(x) = (1/2)^x | 1/2 | Exponential Decay | For every increase of 1 in x, the function's value is halved (decays). |
| f(x) = 1.05^x | 1.05 | Exponential Growth | For every increase of 1 in x, the function's value increases by 5% (growth). |
| f(x) = 0.95^x | 0.95 | Exponential Decay | For every increase of 1 in x, the function's value decreases by 5% (decay). |
Practical Insights
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Growth Rate: When b > 1, b - 1 represents the growth rate. For example, if b = 1.10, the growth rate is 10% per time period.
-
Decay Rate: When 0 < b < 1, 1 - b represents the decay rate. For example, if b = 0.80, the decay rate is 20% per time period.
Understanding the base helps in modeling and interpreting various real-world phenomena, from population growth and compound interest to radioactive decay and the spread of diseases.
