243 = 3*81
=> 81^x = (3*81)^x + 2
=> 81^x = 3^x * 81^x + 2
=> 81^x(1 - 3^x) = 2
=> (3^x)^4 (1 - 3^x) = 2
"Name "y = 3^x", then we have"
=> y^4 (1 - y) = 2
=> y^5 - y^4 + 2 = 0
"This quintic equation has the simple rational root "y= -1."
"So "(y+1)" is a factor, we divide it away :"
=> (y+1)(y^4-2 y^3+2 y^2-2 y+2) = 0
"It turns out that the remaining quartic equation has no real" "roots. So we have no solution as "y = 3^x > 0" so "y=-1
"does not yield a solution for "x.
"Another way to see that there is no real solution is :"
243^x >= 81^x" for positive "x", so "x" must be negative."
"Now put "x = -y" with "y" positive, then we have"
(1/243)^y + 2 = (1/81)^y
"but "0 <= (1/243)^y <= 1" and "0 <= (1/81)^y <= 1
"So " (1/243)^y + 2 " is always bigger than "(1/81)^y.
