Solve for x over the real numbers:
(2/3)^(2 x-1) = (9/4)^(3 x+2)
Take the natural logarithm of both sides and use the identity log(a^b) = b log(a):
-log(3/2) (2 x-1) = log(9/4) (3 x+2)
Expand out terms of the left hand side:
log(3/2)-2 log(3/2) x = log(9/4) (3 x+2)
Expand out terms of the right hand side:
log(3/2)-2 log(3/2) x = 3 log(9/4) x+2 log(9/4)
Subtract 3 x log(9/4)+log(3/2) from both sides:
(-2 log(3/2)-3 log(9/4)) x = 2 log(9/4)-log(3/2)
Divide both sides by -2 log(3/2)-3 log(9/4):
Answer: |x = (log(3/2)-2 log(9/4))/(2 log(3/2)+3 log(9/4)) = - 3/8
