Final answer:
The base of an exponential function must be positive to ensure that the function provides real outputs for all real inputs. Negative bases can result in undefined or complex values when raised to some powers, which makes the function behavior unpredictable and goes beyond real number operations.
Step-by-step explanation:
The base of an exponential function cannot be negative because such functions are defined to produce real outputs for any real input when dealing with real numbers. If you have a negative base, raising it to certain powers, like rational numbers or irrational numbers, can result in undefined or complex numbers, which go beyond the scope of real number operations.
To illustrate why, consider that the exponential and natural logarithm are inverse functions. For example, In (ex) = x and eln(x) = x, these properties hold true because the base 'e' is positive. Distinct characteristics of exponential functions, such as continuous growth or decay, are consistent because the base is positive.
In cases where we talk about exponential growth, such as doubling, we multiply by the base at each interval. For example, after 5 doubling intervals, we have 25 = 32. This formula's consistency is also dependent on the base being positive, as negative bases would introduce irregular oscillations between positive and negative values, defying the expected smooth and predictable pattern of exponential growth.