1.4: Non-Natural Powers

Having defined the exponential and logarithm, we have the tools needed to address the issue raised earlier, i.e. how to define non-natural power operations. First, observe that \[\textrm{For}\;\,y \in \mathbb{N}, \;\;\;\ln(x^y) = \underbrace{\ln(x)\ln(x)\cdots\ln(x)}_{y\;\text{times}} = y \ln(x).\] Hence, by applying the exponential to each side of the above equation, \[x^y = \exp[y \ln(x)] \quad \mathrm{for} \;\,y \in \mathbb{N}.\] We can generalize the above equation so that it holds for any positive \(x\) and real \(y\), not just \(y \in \mathbb{N}\). In other words, we treat this as our definition of the power operation for non-natural powers: \[x^y \equiv \exp[y \ln(x)] \quad\; \mathrm{for}\;\, x \in \mathbb{R}^+, \;y \notin \mathbb{N}.\] By this definition, the power operation always gives a positive result. And for \(y \in \mathbb{N}\), the result of the formula is consistent with the standard definition based on multiplying \(x\) by itself \(y\) times.

This generalization of the power operation leads to several important consequences:

1. The zeroth power yield unity: \[\displaystyle x^0 = 1 \;\;\mathrm{for}\;\, x \in \mathbb{R}^+.\]

2. Negative powers are reciprocals: \[x^{-y} = \exp[-y\ln(x)] = \exp[-\ln(x^y)] = \frac{1}{x^y}.\]

3. The output of the exponential function is equivalent to a power operation: \[\exp(y) = e^y\] where \[e \equiv \exp(1) = 2.718281828459\!\dots\] (This follows by plugging in \(x=e\) and using the fact that \(\ln(e) = 1\).)

4. For \(x \le 0\), the meaning of \(x^y\) for non-natural \(y\) is ill-defined, since the logarithm does not accept negative inputs.