# How to calculate a decimal power of a number

I wish to calculate a power like

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

This follows from the fact that exponentials and logarithms are inverses of each other, and that the logarithm has the property that

So you have, for example,

Or more generally,

In fact, this is formula can be taken as the definition of $a^b$ for $a\gt 0$ and arbitrary exponent $b$ (that is, not an integer, not a rational).

As to computing $e^{2.14\ln(2.14)}$, there are reasonably good methods for approximating numbers like $\ln(2.14)$, and numbers like $e^r$ (e.g., Taylor polynomials or other methods).

Attribution
Source : Link , Question Author : Mixxiphoid , Answer Author : Ross Millikan