#1**0 **

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)"**

Guest Oct 15, 2017

edited by
Guest
Oct 15, 2017