Question

why exponential function can't be zero

Answer 1

The base b in an exponential function must be positive. Because we only work with positive bases, bx is always positive. The values of f(x) , therefore, are either always positive or always negative, depending on the sign of a . ... If b > 1 , the function grows as x increases.


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Answer 2

In Real Analysis, the base of an exponential is defined for all positive real values excluding 11 (1x1x simply equals 11 for all xx).

To answer the question of why the base of an exponential function cannot be negative, consider the function f(x)=(−4)xf(x)=(−4)x.

For integer values of x, our function is well defined. For example, (−4)2=16,(−4)−3=−164(−4)2=16,(−4)−3=−164 and so on.

The real problem arises when we extend x to the entire real number line. The co-domain of our original function becomes complex for certain values, which is not permitted in Real Analysis. For example, if we let x=1/2x=1/2, we have (−4)1/2=√−4=2i.(−4)1/2=√−4=2i. Within the framework of complex analysis, where the entire complex field is our playing ground, such a result would be perfectly acceptable. In Real Analysis, we restrict ourselves to the real number line.