# Thread: Show that x =

1. ## Show that x =

Hi,

My uncertainty lies in the approach to the question. Anyway, I made an attempt.

Given $\displaystyle \displaystyle e^x - e^{-x} = 4$, show that $\displaystyle \displaystyle x = \ln(2+\sqrt{5})$

$\displaystyle \displaystyle \ln e^x - \ln e^{-x} = \ln 4$

$\displaystyle \displaystyle x - (-x) = \ln 4$

$\displaystyle \displaystyle 2x = \ln 4$

$\displaystyle \displaystyle x = \ln 2$

2. $\displaystyle e^x - e^{ - x} = 4\; \equiv \,e^{2x} - 4e^x - 1=0$.

3. $\displaystyle \displaystyle\ln(e^x/e^{-x})=\ln e^x - \ln e^{-x}$, but $\displaystyle \displaystyle\ln(e^x-e^{-x})\ne\ln e^x - \ln e^{-x}$.

4. How did you get this part: $\displaystyle e^{2x} - 4e^x - 1=0?$
Why did you set it as equal to zero?
Is there any other way of doing this?

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$\displaystyle \displaystyle e^{2x} - 4e^x - 1=0$

Let $\displaystyle e^x = y$

$\displaystyle \displaystyle= y^2 - y - 1 = 0$

$\displaystyle \displaystlye= (y - \frac{1}{2})^2 - \frac{5}{4} = 0$

$\displaystyle y = \frac{1}{2}(1 - \sqrt{5})$

$\displaystyle y = \frac{1}{2}(1 + \sqrt{5})$

Resubstituting $\displaystyle y$ for $\displaystyle e^x$:

$\displaystyle e^x = \frac{1}{2}(1 + \sqrt{5})$

$\displaystyle \ln e^x = \ln \frac{1}{2}(1 + \sqrt{5})$

$\displaystyle x = \ln \frac{1}{2}(1 + \sqrt{5})$

OR

$\displaystyle e^x = \ln \frac{1}{2}(1 - \sqrt{5})$

$\displaystyle \ln e^x = \ln \frac{1}{2}(1 - \sqrt{5})$

$\displaystyle x = \ln \frac{1}{2}(1 - \sqrt{5})$

This still isn't it.

5. $\displaystyle e^x - e^{-x} = 4$

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

$\displaystyle e^{2x} - 1 = 4e^x$

$\displaystyle e^{2x} - 4e^x - 1 = 0$

quadratic formula (remember $\displaystyle e^x > 0$ for all $\displaystyle x$)

$\displaystyle e^x = 2 + \sqrt{5}$

$\displaystyle x = \ln(2 + \sqrt{5})$

6. Thanks.

What's the reason for multiplying throughout by e^x?
No other way of doing this?

7. Originally Posted by Hellbent
What's the reason for multiplying throughout by e^x?
No other way of doing this?
It is done to make the $\displaystyle e^x$ in the numerator

Otherwise $\displaystyle e^x - e^{ - x} - 4 = e^x - \frac{1}{e^{x}}- 4$

8. Originally Posted by Hellbent
Thanks.

What's the reason for multiplying throughout by e^x?
No other way of doing this?
to make the equation quadratic in $\displaystyle e^x$ ... and subsequently easier to solve.

9. Originally Posted by skeeter
to make the equation quadratic in $\displaystyle e^x$ ... and subsequently easier to solve.
Thanks, I see it now