1.4: Non-Natural Powers
( \newcommand{\kernel}{\mathrm{null}\,}\)
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 Fory∈N,ln(xy)=ln(x)ln(x)⋯ln(x)⏟ytimes=yln(x). Hence, by applying the exponential to each side of the above equation, xy=exp[yln(x)]fory∈N. We can generalize the above equation so that it holds for any positive x and real y, not just y∈N. In other words, we treat this as our definition of the power operation for non-natural powers: xy≡exp[yln(x)]forx∈R+,y∉N. By this definition, the power operation always gives a positive result. And for y∈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: x0=1forx∈R+.
- Negative powers are reciprocals: x−y=exp[−yln(x)]=exp[−ln(xy)]=1xy.
- The output of the exponential function is equivalent to a power operation: exp(y)=ey where e≡exp(1)=2.718281828459… (This follows by plugging in x=e and using the fact that ln(e)=1.)
- For x≤0, the meaning of xy for non-natural y is ill-defined, since the logarithm does not accept negative inputs.