But that’s hardly exhaustive. Here’s another thing I dig about e, an easy fact for any first-year calculus student to verify:

Also, an addendum: a fun observation from Twitter…

My former student Brandon Coya discovered this "commutative law". It can't be new. It's easy to prove. But I'll name it after him. 🙃@CreeepyJoe quickly noticed that the operation x#y = x^{ln y} is also associative and obeys e#x = x.

Thank you. Here are two more properties that make e unique among the reals. (Does unique ==> natural? Discuss among yourselves.)

(1) e is the positive real number with the greatest “self-root.” When posed as the problem, “Determine the maximum value of f(x) = x^(1/x),” this fact is the correct answer to a standard Calculus I exercise.
(2) The second fact is closely related to your yummy “naturalness” justification. Call two positive real numbers a and b _exponential B.F.F.s_ iff a ≠ b and a^b = b^a. Then e is the unique real number > 1 without an exponential B.F.F.

x#y is an operation I discovered on my own back when I was taking a class on ring/field theory. Some other properties:

1#x = 1^ln x = 1; x#1 = x^(ln 1) = x^0 = 1; so 1 behaves like 0 does in multiplication.

x#(e^(1/ln x)) = e, meaning e^(1/ln x) is the inverse of x (though it’s undefined if x = 1).

(xy)#z = (xy)^ln z = (x^ln z)(y^ln z) = (x#z)(y#z); that is, # distributes over multiplication just like multiplication distributes over addition.

The positive real numbers with the operations × and # are a field, and that field is isomorphic to the real numbers with the usual addition and multiplication.

Doesn’t Coya’s law hold for any logarithm, not just log base e? (except the unit law has e replaced by the base of the log.)

Good point!

Thank you. Here are two more properties that make e unique among the reals. (Does unique ==> natural? Discuss among yourselves.)

(1) e is the positive real number with the greatest “self-root.” When posed as the problem, “Determine the maximum value of f(x) = x^(1/x),” this fact is the correct answer to a standard Calculus I exercise.

(2) The second fact is closely related to your yummy “naturalness” justification. Call two positive real numbers a and b _exponential B.F.F.s_ iff a ≠ b and a^b = b^a. Then e is the unique real number > 1 without an exponential B.F.F.

x#y is an operation I discovered on my own back when I was taking a class on ring/field theory. Some other properties:

1#x = 1^ln x = 1; x#1 = x^(ln 1) = x^0 = 1; so 1 behaves like 0 does in multiplication.

x#(e^(1/ln x)) = e, meaning e^(1/ln x) is the inverse of x (though it’s undefined if x = 1).

(xy)#z = (xy)^ln z = (x^ln z)(y^ln z) = (x#z)(y#z); that is, # distributes over multiplication just like multiplication distributes over addition.

The positive real numbers with the operations × and # are a field, and that field is isomorphic to the real numbers with the usual addition and multiplication.

For any x the tangent at (x, e^x) cuts the x axis at x – 1

This law has been known for a while. A quick search provides a result 4 years old and I would guess a more thorough search would find one much, much older.

https://www.quora.com/Why-is-the-graph-of-x-ln-y-y-ln-x-the-open-first-quadrant-What-class-should-I-have-learned-that-in-and-how-could-I-prove-it