I wish to calculate a power like 2.142.14
When I ask my calculator to do it, I just get an answer, but I want to see the calculation.
So my question is, how to calculate this with a pen, paper and a bunch of brains.
Answer
For positive bases a, you have the general rule
ab=exp(bln(a))=eblna.
This follows from the fact that exponentials and logarithms are inverses of each other, and that the logarithm has the property that
ln(xr)=rln(x).
So you have, for example,
(2.14)2.14=eln((2.14)2.14)(because elnx=x)=e(2.14)ln(2.14)(because ln(xr)=rlnx)
Or more generally,
ab=eln(ab)=eblna.
In fact, this is formula can be taken as the definition of ab for a>0 and arbitrary exponent b (that is, not an integer, not a rational).
As to computing e2.14ln(2.14), there are reasonably good methods for approximating numbers like ln(2.14), and numbers like er (e.g., Taylor polynomials or other methods).
Attribution
Source : Link , Question Author : Mixxiphoid , Answer Author : Ross Millikan