Which is greater? a^b or b^a?

[greymatters]

Here’s an unusual challenge for mental calculation, and I’ve made it to a point where I could use some help from the mental math calculation in this forum.

The challenge is: Given two different positive real numbers, a and b, quickly determine in your head which is greater, a^b (a to the power of b) or b^a (b to the power of a). Ideally, this should be done without mentally calculating the answers themselves.

Having done some work and research on this on my own (links below), I’ve found 3 simple patterns:
Which is Greater: a^b or b^a?
e^pi vs pi^e

1. If both numbers are equal to or greater than e (2.71828…), the smaller number to the power of the larger number (such as 3^4) will be greater than the larger number to the smaller power (such as 4^3).

2. If both numbers are equal to or less than e (2.71828…), the larger number to the power of the smaller number (such as 2^1.5) will be greater than the larger number to the smaller power (such as 1.5^2).

3. If either one of the numbers is e (2.71828…) itself, then e^b will be greater than b^e, regardless of which one is larger.

This just leaves one variation challenge to which I haven’t found a simple rule. What do you do when given two different positive real numbers, where a is less than e and the b is greater than e?

I’ve put together a tool in Desmos, Above/Below e, to work out when the smaller number to the power of the larger number becomes greater than the larger number to the smaller power, focusing on intervals of 1/10th. The points graphed are where this occurs. For example 2^4.1 is greater than 4.1^2. But once b (the larger number, above e) drops below that point, you have either equality (2^4 = 4^2) or the reverse relationship (2^3.9 is less than 3.9^2).

Note that the graphed relationship is that of an exponential curve, which can make this difficult.

Is there a simple mental way to work out which is greater via a rule, like the 3 above, when a is less than e and b is greater than e? If not, is there any other simple mental way to work out which would be greater without performing a^b and b^a in your head directly?

[DLyons]

No easy answer I’m afraid. One global result is that 2 <=a < e < b <=4. But even considering only rational solutions, there is a countable infinity in that range.

Call the ratio r, then “if r is slightly greater than 1, we get solutions with a just less than e, and b just greater than e. On the other hand, if r is large, we get a slightly greater than 1, and b slightly greater than r.”

Best you could do is memorize some version of the typical values given at http://www.qbyte.org/puzzles/p048s.html

[TimothyBohdan]

The evaluation of a^b versus b^a can be tested when e is between a and b using the following thumbrules:
The case numbers refers to the attached graph, which shows an Excel spreadsheet with a on the horizontal axis, b on the vertical axis, and the calculated value of a^b/b^a in each cell. When the value of a^b/b^a is >1, then a^b is larger than b^a. When the value of a^b/b^a is <1, then b^a is larger than a^b. The base and exponent are said to dominate in different regions of the graph, as explained in the nine cases listed below. For example, when b is greater than a and the exponent dominates, this means that a>b will be greater than b^a because the bigger number is in the exponent. Conversely, when b is greater than a and the base dominates, this means that b^a will be greater than a^b because the bigger number is in the base.

Case 1 and Case 2 - when both a and b are greater than e, then the exponent dominates regardless.
Case 3 and Case 4 - when both a and b are less than e, then the base dominates regardless.
Case 5 - When b <= e < a but ln(a)/a < ln(b)/b, then the exponent dominates, so b^a is bigger because a is bigger.
Case 6 - When b < e <= a and ln(a)/a > ln(b)/b, then the base dominates, so a^b is bigger because a is bigger.
Case 7 - When a <= e < b but ln(b)/b < ln(a)/a, then the exponent dominates, so a^b is bigger because b is bigger.
Case 8 - When a < e <= b and ln(b)/b > ln(a)/a, then the base dominates, so b^a is bigger because b is bigger.
Case 9 - When one of the numbers (a or b) is equal to e, then the following is true. For any number c, e^c is always greater than c^e. This follows from cases 5-8 when one of the numbers is equal to e.

It is easier to comprehend the above statements visually on the graph. For the cases where e is in between a and b, then the difference between ln(b)/b and ln(a)/a is the determining factor in deciding whether the base or the exponent dominates. In general when ln(b)/b is less than ln(a)/a, then the exponent dominates. When ln(b)/b is greater than ln(a)/a, then the base dominates. The boundary curve between where the base or exponent dominates is shaped like an exponential decay curve. The boxed numbers are those that are really close to 1.000. This helps to identify the boundary curves separating the various cases.

[Zoomy (Ben Pridmore)]

This is an extremely cool discussion, and it makes a very nice graph, too!