Solve for x with exact values

Question

I am super confused how does this step end up with this?

Then this is the working, I dont understand the second step please help me to show the missing step any Law of Logarithms at work here?

Answer 1

In the second step, both sides of the equation are multiplied by $e^{3x}$. In the third step, $2 \cdot e^{3x}$ is subtracted from both sides of the equation. It can be rephrased as a quadratic equation in $y$ where $y = e^{-3x}$ and $y \in \mathbb{R^{+}}$.

Answer 2

Perhaps this might be easier to follow using substitution.

With the substitution, $y=e^{3x}$ (forcing $y>0$), this becomes $$ \begin{align} y-8/y&=2\tag1\\ y^2-2y-8&=0\tag2\\ y&=1\pm3\tag3 \end{align} $$ Explanation:
$\text{(1):}$ substitution
$\text{(2):}$ algebra
$\text{(3):}$ quadratic formula

Therefore, $e^{3x}=4\Rightarrow x=\frac23\log(2)$.