Express cosh 2x and sinh 2x in exponential form; solve for real values of x for the following:

2 cosh 2x - sinh 2x = 2

2 cosh(2x) - sinh(2x) = 2
==> 2 * (1/2)(e^(2x) + e^(-2x)) - (1/2)(e^(2x) - e(-2x)) = 2, by definition of sinh and cosh
==> 2(e^(2x) + e^(-2x)) - (e^(2x) - e(-2x)) = 2 * 2, multiplying both sides by 2.

Simplifying:
e^(2x) + 3 e^(-2x) = 4
==> e^(4x) + 3 = 4e^(2x), multiplying both sides by e^(2x)
==> e^(4x) - 4e^(2x) + 3 = 0

Solve by factoring:
(e^(2x) - 3)(e^(2x) - 1) = 0.
==> e^(2x) = 3 or 1
==> 2x = ln 3 or 0
==> x = (1/2) ln 3 or 0.

I hope this helps!