Solving for y
$\displaystyle y^{2} - 9 = e^{y}$
Is this the way to go?
$\displaystyle (e^{y})(y^{2}) - (e^{y})(9) = e^{y}(e^{y})$
Make everything e to the something and then ln it all away.
The solution should be -3 because that's where the two functions intersect. Maybe it was a number a little less or more than -3, but it was somewhere near there. Actually it can't be -3 cause plugging it into the equations does not make them equal.
$\displaystyle y^{2} - 9 = e^{y}$
$\displaystyle (e^{y})(y^{2}) - (e^{y})(9)} = (e^{y})(e^{y})$
$\displaystyle ye^{y + 2}} - 9e^{y} = e^{2y}$
$\displaystyle \ln[ye^{y + 2} - 9e^{y}] = \ln[e^{2y}}]$
$\displaystyle y + 2 - 9y = 2y$
$\displaystyle -8y + 2 = 2y$
$\displaystyle 2 = 10y$
$\displaystyle y = \dfrac{1}{5}$ ??
Sorry if some big mistake here. Probably there is.
Agreed, because above there was a major algebraic mistake in trying to "make everything e to the something", and then "Ln it all" to normal digits that are easy to work with (ones with no "e" or "Ln" stuff.).You won't be able to get exact solutions for this equation. You will have to use numerical method of solution, like the Bisection Method or Newton's Method.
There is basically nothing about that attempted solution which is correct. Like I said, there is no point trying to solve it analytically when the variable to be solved for is both inside and outside of a function, like it is here.
y = -3 is clearly not a solution, since the LHS is $\displaystyle \begin{align*} (-3)^2 - 9 = 0 \end{align*}$ and the RHS is $\displaystyle \begin{align*} \mathrm{e}^{-3} \neq 0 \end{align*}$. These are clearly not the same thing.
First of all, $\displaystyle \begin{align*} \mathrm{e}^y\,y^2 \neq y\,\mathrm{e}^{y + 2} \end{align*}$, so the first line is invalid.
Taking logarithms of both sides is ok, but $\displaystyle \begin{align*} \ln{ \left( y\,\mathrm{e}^{y + 2} - 9\mathrm{e}^y \right) } \neq y + 2 - 9y \end{align*}$. Go back and review your logarithm laws. $\displaystyle \begin{align*} \log{(a+b)} \neq \log{(a)} + \log{(b)} , m\log{(n)} \neq \log{(mn)} \neq \log{(n)} \end{align*}$. You have essentially tried to simplify using these three fallacies.
$\displaystyle y^{2} - 9 = e^{y}$
$\displaystyle (e^{y})(y^{2}) - (e^{y})(9)} = (e^{y})(e^{y})$
$\displaystyle e^{y}y^{2} - 9e^{y} = e^{2y}$
$\displaystyle \ln[e^{y}y^{2} - 9e^{y}] = \ln[ e^{2y}]$
$\displaystyle \ln[\dfrac{e^{y}y^{2}}{9e^{y}}] = \ln[ e^{2y}]$
??