Log/Exponential problem

**desiderius1** · April 18th 2010, 08:24 PM

$ \frac{e^x-e^{-x}}{2}=10 $

Answer:
$ x = ln (10 + \sqrt{101} ) $

What I'm doing (More than likely wrong)

$ \frac{e^x - e}{2^x} = 10 $

$ e^x-e=10(2^x) $

$ e^x-e=20^x $

**sa-ri-ga-ma** · April 18th 2010, 08:33 PM

According to the answer, the given problem is wrong.
The equation should be equated to 10 rather than 7.

**Failure** · April 18th 2010, 08:37 PM

Originally Posted by desiderius1

$ \frac{e^x-{\color{blue}e^{-x}}}{{\color{red}2}}=7 $

Answer:
$ x = ln (10 + \sqrt{101} ) $

This answer is wrong, assuming that you have managed to write down the correct equation that you were asked to solve.

What I'm doing (More than likely wrong)

$ \frac{e^x - {\color{blue}e}}{{\color{red}2^x}} = 7 $

The above equation is very different from the one that you were given.

$ e^x-e=7(2^x) $
$ e^x-e=14^x $

That last step is not valid. More generally, you have that in general $a\cdot b^n\neq (a\cdot b)^n$ .

**desiderius1** · April 18th 2010, 08:58 PM

Sorry, there are two identical problems, one equating to 10 and the other 7, I misread while typing.

The problem is:
$ \frac{e^x-e^{-x}}{2}=10 $

**Failure** · April 19th 2010, 05:03 AM

Originally Posted by desiderius1

Sorry, there are two identical problems, one equating to 10 and the other 7, I misread while typing.

The problem is:
$ \frac{e^x-e^{-x}}{2}=10 $

The easy way to solve this equation would be to recognize that the left side really is $\sinh(x)$ , and that the solution is simply $x=\arsinh(10)$ .

Or else you can first multiply the equation by $2e^x$ to get a quadratic equation for $e^x$ :

$\frac{e^x-e^{-x}}{2}=10 \qquad | \cdot 2e^x$

$\left(e^x\right)^2-1=20 e^x \qquad | -20e^x$

$\left(e^x\right)^2-20e^x-1=0$
Thus, we get

$\left(e^x\right)_{1,2} = \frac{-(-20)\pm \sqrt{(-20)^2-4\cdot 1\cdot (-1)}}{2\cdot 1}=10\pm \sqrt{101}$
But only the $+$ case is possible, thus we have

$e^x=10+\sqrt{101}\qquad | \ln$

$x=\ln(10+\sqrt{101})$

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