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# e calculation

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e^w+22=21e^-w

Oct 15, 2017

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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

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)"

Oct 15, 2017
edited by Guest  Oct 15, 2017