e^a+e^-a=10

e^a + 1/e^a = 10 : combine the left side into one

(e^a * e^a + 1) / e^a = 10

(e^(2a) + 1) / e^a = 10

e^(2a) + 1 = 10 * e^(a)

e^(2a) - 10 * e^(a) + 1 = 0 : quadratic in e^(a), let z = e^(a)

e^(a) * e^(a) - 10 * e^(a) + 1 = 0

z^2 - 10*z + 1 = 0

z = 5 +/- sqrt(96) / 2 : replace z

e^(a) = 5 +/- 2*sqrt(6)

a = ln( 5 +/- 2 * sqrt(6) )

tada -slim