Solve for x over the real numbers:
e^x-4/e^x = 3
e^x-4/e^x = e^x-4/e^x:
e^x-4/e^x = 3
Simplify and substitute y = e^x:
e^x-4/e^x = e^x-(4)/e^x = y-4/y = 3:
y-4/y = 3
Bring y-4/y together using the common denominator y:
(y^2-4)/y = 3
Multiply both sides by y:
y^2-4 = 3 y
Subtract 3 y from both sides:
y^2-3 y-4 = 0
The left hand side factors into a product with two terms:
(y-4) (y+1) = 0
Split into two equations:
y-4 = 0 or y+1 = 0
Add 4 to both sides:
y = 4 or y+1 = 0
Substitute back for y = e^x:
e^x = 4 or y+1 = 0
Take the natural logarithm of both sides:
x = log(4) or y+1 = 0
Subtract 1 from both sides:
x = log(4) or y = -1
Substitute back for y = e^x:
x = log(4) or e^x = -1
e^x = -1 has no solution since for all z element R, e^z>0 and -1<0:
Answer: |x=2log(2) log=natural log.
