Solve for w over the real numbers:
e^w + 22 = 21/e^w
21/e^w = 21 e^(-w):
e^w + 22 = 21/e^w
Multiply both sides by e^w:
22 e^w + e^(2 w) = 21
Simplify and substitute x = e^w.
22 e^w + e^(2 w) = 22 e^w + (e^w)^2
= x^2 + 22 x:
x^2 + 22 x = 21
Add 121 to both sides:
x^2 + 22 x + 121 = 142
Write the left hand side as a square:
(x + 11)^2 = 142
Take the square root of both sides:
x + 11 = sqrt(142) or x + 11 = -sqrt(142)
Subtract 11 from both sides:
x = sqrt(142) - 11 or x + 11 = -sqrt(142)
Substitute back for x = e^w:
e^w = sqrt(142) - 11 or x + 11 = -sqrt(142)
Take the natural logarithm of both sides:
w = log(sqrt(142) - 11) or x + 11 = -sqrt(142)
Subtract 11 from both sides:
w = log(sqrt(142) - 11) or x = -11 - sqrt(142)
Substitute back for x = e^w:
w = log(sqrt(142) - 11) or e^w = -11 - sqrt(142)
e^w = -11 - sqrt(142) has no solution since for all z element R, e^z>0 and -11 - sqrt(142)<0:
w = log(sqrt(142) - 11) Log here is "natural log(ln)"
