Question

How do you solve `e^x+e^-x=4`?

Answer 1

`x=ln(2+sqrt(3)) or x=ln(2-sqrt(3))`

Explanation:

Temporarily simplify by letting `y =e^x`

`e^x+e^(-x) =4`
becomes `y+1/y = 4`

Multiplying by `y` and sifting things around:
`color(white)("XXXX")y^2-4y+1=0`
Applying the quadratic formula, we get
`color(white)("XXXX")y =2+sqrt(3)color(white)("XXXX")orcolor(white)("XXXX")y=2-sqrt(3)`

If `y = e^x = 2+sqrt(3)`
`color(white)("XXXX")ln(e^x) = x = ln(2+sqrt(3))`

Similarly, if `y=e^x=2-sqrt(3)`
`color(white)("XXXX")x = ln(2-sqrt(3))`