# Thread: [SOLVED] exponential e problem, solve for roots

1. ## [SOLVED] exponential e problem, solve for roots

Hey all, I seem to be having some trouble finding the solutions (roots) to this equation:

0=210-10(e^(x/30) + e^(-x/30))

I got it down to this lol:

21=e^(x/30) + e^(-x/30)

and it seems like it should be easy, just I can't seem to simplify it down enough to be able to take the natural log from both sides

Any help would be greatly appreciated,

thanks!

2. Originally Posted by tkdiamond08
Hey all, I seem to be having some trouble finding the solutions (roots) to this equation:

0=210-10(e^(x/30) + e^(-x/30))

I got it down to this lol:

21=e^(x/30) + e^(-x/30)

and it seems like it should be easy, just I can't seem to simplify it down enough to be able to take the natural log from both sides

Any help would be greatly appreciated,

thanks!
Let $\displaystyle w = e^{x/30}$: $\displaystyle 21 = w + \frac{1}{w}$.

Re-arrange this into a quadratic equation in w, solve for w and hence solve for x.

3. Originally Posted by tkdiamond08
Hey all, I seem to be having some trouble finding the solutions (roots) to this equation:

0=210-10(e^(x/30) + e^(-x/30))

I got it down to this lol:

21=e^(x/30) + e^(-x/30)

and it seems like it should be easy, just I can't seem to simplify it down enough to be able to take the natural log from both sides

Any help would be greatly appreciated,

thanks!
let $\displaystyle u = \frac{x}{30}$

$\displaystyle 21 = e^u + e^{-u}$

multiply every term by $\displaystyle e^u$ ...

$\displaystyle 21e^u = e^{2u} + 1$

$\displaystyle 0 = (e^{u})^2 - 21e^u + 1$

$\displaystyle e^u = \frac{21 \pm \sqrt{(-21)^2 - 4}}{2}$

$\displaystyle e^u = \frac{21 \pm \sqrt{437}}{2}$

$\displaystyle u = \ln\left(\frac{21 \pm \sqrt{437}}{2}\right)$

$\displaystyle x = 30\ln\left(\frac{21 \pm \sqrt{437}}{2}\right)$

4. Thanks guys!

Oh and sorry for initially posting it in the wrong forum.