What Is the Solution to sinh(x) = 1?

[kd001]

Homework Statement

sinh(x) = 1

What is the value of 'x'?

Homework Equations

sinh(x) = (1/2)(e^x - e^-x)

The Attempt at a Solution

e^x - e^-x = 2

Then what do I do?

Thanks

 

[emanuel_hr]

Multiply the whole eq by $$e^{x}$$ then solve the resulting quadtatic eqn in $$e^{x}$$ and afterwards keep the positive solution and take its logarithm to obtain x.

 

[arkajad]

Multiply the whole eq by $$e^{x}$$ then solve the resulting quadtatic eqn in $$e^{x}$$

That is put, say, $$u=e^x$$ and solve for u. Then find x from $$e^x=u$$.

 

[kd001]

Thanks. That worked.

 

[HallsofIvy]

Homework Statement

sinh(x) = 1

What is the value of 'x'?

Homework Equations

sinh(x) = (1/2)(e^x - e^-x)

The Attempt at a Solution

e^x - e^-x = 2

Then what do I do?

Thanks

Multiply on both sides by $e^x$ to get $(e^x)^2- 1= 2e^x$ and then subtract $2e^x- 1$ from both sides: $(e^x)^2- 2e^x= 1$. Think of that as a quadratic equation in $e^x$ and complete the square: $(e^x)^2- 2e^x+ 1= (e^x- 1)^2= 2$.

Take the square root of both sides, $e^x- 1= \pm\sqrt{2}$ and add 1 to both sides, $e^x= 1\pm\sqrt{2}$. $1- \sqrt{2}< 0$ so you get the single solution $x= ln(1+ \sqrt{2})$.

Too slow! Too slow!