How do I solve sinh(x)=1

You need to remember that `(e^x - e^(-x))/2`  gives
the hyperbolic sine, hence, you need to solve the equation `(e^x - e^(-x))/2 = 1`  such
that:

`(e^x - e^(-x))/2 = 1 =gt (e^x - e^(-x)) =
2`

You need to write `e^(-x)`  making use of negative power
property such that:

`e^(-x) =
1/e^x`

`e^x - 1/e^x - 2 =
0`

You need to bring the terms to a common denominator such
that:

`e^(2x) - 2e^x - 1 =
0`

You should come up with the substitution`e^x = t`  such
that:

`t^2 - 2t - 1 =
0`

`t_(1,2) =
(2+-sqrt(4+4))/2`

`t_1 = (2+2sqrt2)/2 =gt t_1 =
1+sqrt2`

`t_2 = 1-sqrt2`

You
need to solve for x the equations `e^x = t_1`  and `e^x = t^2`  such
that:

`e^x = 1+sqrt2 =gt x =
ln(1+sqrt2)`

`e^x = 1-sqrt2 lt 0`  impossible because
`e^x`  needs to be strictly positive.

Hence,
evaluating the solution to equation sinh = 1 yields `x = ln(1+sqrt2).`