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I need help solving $2\mathrm{e}^x=5-\mathrm{e}^{-x}$.

I've tried many ways of solving it but I keep getting the wrong answer.

By the way, my book says the solutions are $x=-1.518$ and $x=0.825$

Thanks!

Fly by Night
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Xerif917
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  • The answers, rounded to 3.d.p., will be $-1.518$ and $0.825$ but they are not the exact answers. In fact, the exact answers will be expressed in terms of logarithms. For example the answer which rounds to $0.825$ is in fact

    $$0.82451591412420971886\ldots$$

    Where the number carries on for ever. Imagine you had a job and your salary was $44,! 567.89$ a year and your boss gave you $44,! 000$ saying that was the same as $44,! 567.89$; would you accept it?

     – Fly by Night Apr 16 '14 at 22:58

1 Answers1

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Hint: Let $y=e^x$. Our equation becomes $2y=5-\dfrac{1}{y}$. With manipulation this becomes a quadratic equation in $y$. Solve for $y$. Then find $x$.

André Nicolas
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