which matters more; the base or the exponent?

[4=5]

if you take two numbers like two and three and exponentiate them 3^2 > 2^3
In this case, putting the larger number as the base wins.
But as you move to larger pairs of numbers the exponent takes more control over the result to the point where the base hardly matters.

So if we put them right at the point where both are equally important like in 2^4 = 4^2
What is the size of the difference between the two chosen numbers as we increase the smaller of the two? (limiting ourselves to real numbers larger than 1)


I graphed x^(x+2^-y) = (x+2^-y)^x to have a look,
and it seems to have some very funny things about it

when x=e y spikes up to around 16
when x=4 y jogs towards the x-axis a step
and again when x=8
and again when x=16

Is there a good reason for the step shape at every power of two or is that just a rounding error?
I find it funny that out in the humongous numbers, x^(x+1) = (x+1)^x

[tomtom2357]

I find it funny that out in the humongous numbers, x^(x+1) = (x+1)^x

This is not true. x^x+1=x*x^x, but (x+1)^x=x^x*(1+(1/x))^x which approaches e*x^x, so x^x+1 grows faster than (x+1)^x. Doing some partial derivatives, I found that for x^y, y matters more when y<x ln(x), and x matters more when y>x ln(x), and they contribute the same when y=x ln(x), which explains the spike on your graph at e, because at the point (e,e) x and y matter the same amount.

[fishfry]

But as you move to larger pairs of numbers the exponent takes more control over the result to the point where the base hardly matters.

The base doesn't matter much because all bases are essentially the same. We have

a^b = e^b*log(a)

Since log(a) increases slowly with a, you can see that what really drives the growth of a^b is b.

That's the fundamental explanation for what you observe: The growth of the exponent has a far larger impact than the growth of the base.