A typical example would be a=2 and b=4; 2^4=4^2=16. The suggested problem is to construct a GLOBAL CHARACTERIZATION; algebraic, computational, geometric, and otherwise, that will determine ALL such positive real number pairs (a,b).

Possible Course of Action:

1. Transform the given algebraic relationship a^b= b^a into an equivalent logarithmic relation to get initial clues as to how you might proceed with the problem.
2. Based on this logarithmic relationship, construct a related "generic" function f(x) in terms of a single variable "x" to assist you to see what the connection between the two numbers ought to be.
3. Draw a graph of the relationship found/proposed in step 2 on TWIDDLE or PLOT to see the geometric connection that has to hold for such number pairs.
4. Construct a simple procedure, an ALGORITHM, to compute such number pairs on a pocket calculator (no computer programs please). Describe your procedure with examples (numerical/otherwise). You might consider building a recursive scheme that will seek the "other" positive number (with a prescribed resolution) when the user picks the first number.
5. Relevant Questions: Are there positive integer pairs other than the one cited in the above example (a=2,b=4) with the said property? If so, can you identify all such integer pairs? Are there rational pairs with this property? Are there irrational pairs with the same property? Can you come up with a clever scheme for finding ALL such rational/irrational pairs with any desired level of accuracy ?

Graph of the generic function f(x) constructed in step3 should shed light on most of these questions, at least at an intuitive level. Can you think of additional graphs at this point that may take you to the "inner secrets" of this problem? If so, try drawing such graphs on TWIDDLE to look at the problem from different angles. Do you see the number "e" somehow getting into the picture?