Algebra AND Logarithm - Interesting Question!

[@yurenchu]

3ˣ - 3ʸ = 3
3ˣ⁺ʸ = 3
Substitute u = 3ˣ , v = 3ʸ . Note: since x and y are real, u and v must be real and positive.
u - v = 3
uv = 3
u = 3/v ==> substitute into first equation
3/v - v = 3
3 - v² = 3v
3 = v² + 3v
3 + 9/4 = v² + 2(3/2)v + (3/2)²
21/4 = (v + 3/2)²
v + 3/2 = ±√(21/4)
v = -3/2 ± (1/2)√21
v = (-3 - √21)/2 OR v = (-3 + √21)/2
v = (-3 - √21)/2 < 0 ==> no real solution for (x,y)
v = (-3 + √21)/2 > 0 ==>
... u - v = 3 ==> u = v+3 = v + 6/2 ...
u = (3 + √21)/2
x = ln(u)/ln(3) = ln( (3 + √21)/2 )/ln(3)
= [ ln(3 + √21) - ln(2) ]/ln(3)
y = ln(v)/ln(3) = ln( (-3 + √21)/2 )/ln(3)
= [ ln(-3 + √21) - ln(2) ]/ln(3)

[@Mathsandolympiad]

Thanks