Inferring the value from the set of pairwise multiplication

Question

I am not so familiar with the literature of math. Therefore I could not find out if it is something in the literature or how I can search for it.

In my problem, I have 5 different random real numbers, namely $a, b, c, d, e$. I am given the pairwise multiplication of some of these values in a set $S$ where $S = \{ab, ac, ae, de\}$. The question is that based on the set $S$, can you infer the value of any of these variables? In general, how can you prove whether you can obtain or not the variable based on the given set of pairwise multiplication of some those variables?

P.S: I am not totally sure if I used the correct tags. I would appreciate your help about it. Thanks in advance.

Answer 1

Hint:

Take the logarithms of all quantities to turn the products in sums ($\log ab=\log a+\log b$). Then the ocean of linear algebra is yours.

Answer 2

For a given set of values $A,B,C,D$, the set of four equations for five variables $$ab=A\\ac=B\\ae=C\\de=D$$

will not have a unique solution. For example, if $A,B,C,D$ are all nonzero, then for every value of $b$, you can choose $$a=\frac{A}{B}\\c=\frac{B}{A}b\\d=\frac{DA}{Cb}\\e=\frac{C}{A}b$$

and your conditions will be met.

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In particular, for every value of $b$, setting

$$a=\frac1b\\ c=2b\\d=\frac{4}{3b}\\ e=3b$$

will ensure that $S=\{1,2,3,4\}$, but for different values of $b$, you can get completely different values of $a,b,c,d,e$.