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AB or BA

A simple question this week. Given two real numbers: A and B, which generates the larger number? AB or BA ?
Is AB > BA or BA > AB ?
Is it better to use the larger number as the base or the exponent? It seems such a trivial thing, even if we’re not sure, how hard can it be to test a few examples?
1.66.2 < 6.21.6  but   1.66.3 > 6.31.6
Somewhere in between {1.6,6.2–6.3} it flips over, and also somewhere {1.6–1.7,6.2}, so the transition point depends on both numbers.
Something is clearly going on. Want to experiment yourself to see if you can see a pattern?

Try it out yourself

Below is a little app for you to be able to experiment looking for a pattern. Be careful, using powers, it’s pretty easy to cause overflows. You can also use negative numbers, but bear in mind that negative numbers with non-integer exponents are not defined.


Did you find a pattern? It's not obvious.
Let’s assume that from the set {A,B} that the equality holds
We can raise both sides to the same power without affecting the inequality. Let’s raise each side by the power 1/(AB).
So we can see that the inequality holds if the xth root of x of one number is larger than the other.


Here's a graph of the function:
It's an interesting curve. Eventually it asymptotes to one.
To find out which number to put on the exponent, and which the base, simply look up both numbers on the x-axis and find their value on the curve. The number that represents the higher value when read on the y-axis is the number to put in the base.
We can use calculus to find the maximum value of the curve. (Differentiating and setting this to zero to find the turning point). This occurs when x=e.
This tells us all we need to know to answer the question.

Additional Trivia

We can use the curve above to answer questions about what pairs of numbers give the same result when raised to each others' power.
When x>1 then there is always a chiral value that completes the pair xy=yx.
For example 1.219.5763 = 19.57631.2
As regards integer solutions to xy=yx, there is only one, and this is 24=42.