Simplification!
27^x=(3^2)^2x+1
Convert to base 3!
(3^3)x
Exponent rules!
3^3x
Exponent rules, electric boogaloo!
3^3x=(3^2)^2x+1
uhh...
Derivative junk! a^f(x)=a^g(x) then f(x)=g(x)
Solve it!
x=-2
And you wanted to figure out what that junk is? Uhhh.
*indescribable mumbling originating from Varxaax's mouth*
1/729
yay!