if a power b=b power a then prove a=b

Question

if a power b=b power a then prove a=b

 

Answer 1

2⁴=4²=16 and 2≠4

Answer 2

Okay, given A^B= B^A

Let's define that A >= B, so if they're ever different, A is the bigger one.

Let's define a constant K such that A = KB, then we'll try to prove that K must equal 1.

A^B = B^A

We'll substitute our identity in the right side exponent

A^B = B^(KB)

Well take the Bth root of both sides

A = B^K

We'll substitute our identity in the left side

KB = B^K

and divide by B

K = B^(K-1)

 

Hmmm.  I don't see a way to prove that impossible. 

So, let's substitute simple answers here and see if we're getting anywhere-

if K = 2, then what are B and A?

2 = B^(2-1) = B = 2

A = K B = 2*2 = 4

4^2 = 2^4 ?   16 = 16 check 

 

THEORY DISPROVEN!.

 

if K = 3, then what are B and A?

3 = B^(3-1) = B^2  then B = sqrt(3)

A = KB = 3 sqrt(3)

(3 sqrt(3))^(sqrt(3)) = sqrt(3)  ^ (3sqrt(3)) ?

here I resort to a calculator, 17.362 = 17.362 check.

 

THEORY DISPROVEN!.

 

There's a whole family of counterexamples. 

 

By the way, you don't need to do any Ks under 1, because you'll just get the same pair in a different order.  For instance, k = 1/2 will get you the pair 2,4 again.