Solve sinh(x) = 1 for x?

(e^x - e^(-x))/2 = 1 =>
e^x - e^(-x) = 2 =>
e^x - 2 - e^(-x) = 0 =>
[e^x]^2 - 2(e^x) - 1 = 0.
Let u = e^x, then
u^2 - 2u - 1 = 0 =>
u = 1 +/- (1/2)*sqrt(8) = 1 +/- sqrt(2).
Because e^x cannot be negative, only the "plus" solution makes sense:
e^x = 1 + sqrt(2) =>
x = ln[1 + sqrt(2)] = about 0.88137.

Check:
Google calculator says
sinh(0.88137) = 0.99999493. Swell!

sinh(x) = (1/2) * (e^(x) - e^(-x))

sinh(x) = 1
(1/2) * (e^(x) - e^(-x)) = 1
e^(x) - e^(-x) = 2
e^(2x) - 1 = 2 * e^(x)
e^(2x) - 2 * e^(x) = 1
e^(2x) - 2 * e^(x) + 1 = 2
(e^(x) - 1)^2 = 2
e^(x) - 1 = +/- sqrt(2)
e^(x) = 1 +/- sqrt(2)

1 - sqrt(2) < 0, so we can omit that

e^(x) = 1 + sqrt(2)
x = ln(1 + sqrt(2))