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:
-
The zeroth power yield unity: \[\displaystyle x^0 = 1 \;\;\mathrm{for}\;\, x \in \mathbb{R}^+.\]
-
Negative powers are reciprocals: \[x^{-y} = \exp[-y\ln(x)] = \exp[-\ln(x^y)] = \frac{1}{x^y}.\]
-
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
\(\ln(e) = 1\)
.)
-
For
\(x \le 0\)
, the meaning of
\(x^y\)
for non-natural
\(y\)
is ill-defined, since the logarithm does not accept negative inputs.