What contributes more to a increase in value of a exponentiation: a increase of base or exponent?

Question

Firstly, my apologies for being a programmer messing in math land.

I was wondering whether the value of a exponentiation is increased more by a increase in the base or exponent (I believe this is a kind of limit?), and how this depends on the concrete values of the base and exponent. I wrote a program that, given a base, calculates the value of the exponent such that a equal increase in either results in the same final value for the exponentiation.

def binary_search(under, over, isOver):
    while abs(under-over)>0.00000000001:
        #print under, over
        middle= (under+over)/2
        if isOver(middle):
            over= middle
        else:
            under=middle
    return under

base= 2
small=0.000000000000001
for base in range(2,10):
    base= float(base)
    print binary_search(0.0,100.0, lambda x: (base+small)**x>base**(x+small))

For the bases from 2 to 10, here are the results:

1.87390832634
3.44985378173
6.11740351597
16.0
16.0
16.0
16.0
16.0

Now, obviously, I don't know what I'm doing. I just wanted to know if there's some significance to the convergence to 16 or I'm just doing something horribly wrong (seems likely).

Answer 1

I suspect you're running into floating point inaccuracies. How accurately does your system represent numbers like 50.000000000000001? There should be no need to work with such a tiny value of "small".

In any case, what you are after is this: Given a value $b$, at what value of $exp$ are the partial derivatives of the function $f(base,exp)=base^{exp}$ equal to each other, given that the derivatives are evaluated at $base=b$.

The partial derivative of $f$ with respect to $base$ (which is the rate of increase of $f$ per increase in the value of $base$) is $base^{exp-1}\cdot exp$, and the partial with respect to $exp$ (which is the rate of increase of $f$ per increase in the value of $base$) is $base^{exp}\ln(base)$. Given $base=b$, these are equal when $x=b\ln(b)$.

The values I think you should have gotten for bases 2 through 10 are 1.38629, 3.29584, 5.54518, 8.04719, 10.7506, 13.6214, 16.6355, 19.775, and 23.0259. (Thanks, Mathematica.)

If you compare $4^{5.54518+.001}$ and $(4+.001)^{5.54518}$, you'll find they're closer together than $4^{6.11740+.001}$ and $(4+.001)^{6.11740}$ (using the answer you got).