1. Log/Exponential problem

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

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

What I'm doing (More than likely wrong)

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

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

$\displaystyle e^x-e=20^x$

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

3. Originally Posted by desiderius1
$\displaystyle \frac{e^x-{\color{blue}e^{-x}}}{{\color{red}2}}=7$

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

$\displaystyle \frac{e^x - {\color{blue}e}}{{\color{red}2^x}} = 7$
The above equation is very different from the one that you were given.

$\displaystyle e^x-e=7(2^x)$
$\displaystyle e^x-e=14^x$
That last step is not valid. More generally, you have that in general $\displaystyle a\cdot b^n\neq (a\cdot b)^n$.

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

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

5. 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:
$\displaystyle \frac{e^x-e^{-x}}{2}=10$
The easy way to solve this equation would be to recognize that the left side really is $\displaystyle \sinh(x)$, and that the solution is simply $\displaystyle x=\arsinh(10)$.

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

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

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

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

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

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

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