Math question

How do you solve this equation? 

$\dfrac{1}{e^x - e^{-x}}= 2$

$\begin{array}{rcl} \dfrac{1}{e^x - e^{-x}} &=& 2 \\ e^x - e^{-x} &=& \dfrac{1}{2} \qquad | \qquad \dfrac{e^x - e^{-x}}{2} = \sinh{(x)}\\ 2\sinh{(x)} &=& \dfrac{1}{2} \\ \sinh{(x)} &=& \dfrac{1}{4}\\ x &=& \text{arsinh}{ \left(\dfrac{1}{4} \right)} = \text{sinh}^{-1}{ \left(\dfrac{1}{4} \right)}\\ \mathbf{x} & \mathbf{=} & \mathbf{ 0.2474664615472692 } \end{array}$

Come on Namadesto,

We have answered 2 of these questions for you in the last 12 hours.

How about you giving us some of your thoughts on how you might be able to solve it ?

We can talk those through with you if you like :)

Hint:  let y=e^x

You can simplify before or after you do the substitution.  :)

(1/(e^x - e^-x)) = 2   

Notice that we can invert both sides and get :

(e^x - e^-x)  = 1/2     multiply through by 2e^x  

2e^2x - 2 = e^x      subtract e^x from both sides and rearrange

2e^2x - e^x - 2  = 0    this will not factor......let e^x  = y   and we have

2y^2 - y - 2 = 0     and the solutions for this are....y ≈ -.78078     and y  = 1.2808

Therefore, e^x = -.78078 (impossible).......  or e^x = 1.2808

So

e^x  =  1.2808     which means that x = ln 1.2808  = about .2475