Is it mathematically possible to solve this problem? $$\vec{y}= \log{\frac{\vec{x}}{\sqrt[3]{x_1 \cdot x_2 \cdot x_3}}}$$
Here, $x$ and $y$ are always vectors of length 8 and all elements of $x$ are greater than $0$. $x_1$, $x_2$, and $x_3$ are the first three elements of $x$; is it mathematically possible to determine these elements given you only have $y$?  I should note that $x$ is known for some of my dataset, which could be useful to find a solution for the other portion of the data where $x$ is unknown.
We have
$$x_i=\sqrt[3]{x_1x_2x_3}\,e^{y_i}.$$
Multiplying the equations for $i=1,2,3$, you get the compatibility condition
$$1=e^{y_1+y_2+y_3}.$$ Without it, no solution exists.
Then using  ratios,
$$x_2=x_1e^{y_2-y_1}$$
$$x_3=x_1e^{y_3-y_1}$$ and you are free to choose $x_1$.
I think what you are asking is that for vectors:
$$y=\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix},x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}$$
then is there a way to calculate:
$$y=\begin{pmatrix}\log\frac{x_1}{(\prod x)^{1/3}}\\\log\frac{x_2}{(\prod x)^{1/3}}\\\log\frac{x_3}{(\prod x)^{1/3}}\end{pmatrix}$$
notice that:
$$y_1+y_2+y_3=\log(x_1x_2x_3)-3\log(\prod x)^{1/3}=\log(\prod x)-\log(\prod x)=0$$
and since we know $|y|=8$ we have the system:
$$y_1+y_2+y_3=0$$
$$y_1^2+y_2^2+y_3^2=64$$