Solve for x over the real numbers:
(5+e^x)/(4+9/e^x) = 2
Simplify and substitute y = e^x:
(5+e^x)/(4+9/e^x) = (e^x (5+e^x))/(9+4 e^x) = (y (y+5))/(4 y+9) = 2:
(y (y+5))/(4 y+9) = 2
Multiply both sides by 4 y+9:
y (y+5) = 2 (4 y+9)
Expand out terms of the left hand side:
y^2+5 y = 2 (4 y+9)
Expand out terms of the right hand side:
y^2+5 y = 8 y+18
Subtract 8 y+18 from both sides:
y^2-3 y-18 = 0
The left hand side factors into a product with two terms:
(y-6) (y+3) = 0
Split into two equations:
y-6 = 0 or y+3 = 0
Add 6 to both sides:
y = 6 or y+3 = 0
Substitute back for y = e^x:
e^x = 6 or y+3 = 0
Take the natural logarithm of both sides:
x = log(6) or y+3 = 0
Subtract 3 from both sides:
x = log(6) or y = -3
Substitute back for y = e^x:
x = log(6) or e^x = -3
e^x = -3 has no solution since for all z element R, e^z>0 and -3<0:
Answer: |x = log(6)
