Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

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 ForyN,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)]foryN. We can generalize the above equation so that it holds for any positive x and real y, not just yN. In other words, we treat this as our definition of the power operation for non-natural powers: xyexp[yln(x)]forxR+,yN. By this definition, the power operation always gives a positive result. And for yN, 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: x0=1forxR+.

  2. Negative powers are reciprocals: xy=exp[yln(x)]=exp[ln(xy)]=1xy.

  3. The output of the exponential function is equivalent to a power operation: exp(y)=ey where eexp(1)=2.718281828459 (This follows by plugging in x=e and using the fact that ln(e)=1.)

  4. For x0, the meaning of xy for non-natural y is ill-defined, since the logarithm does not accept negative inputs.


This page titled 1.4: Non-Natural Powers is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?