How to solve 3 variables problem with logarithm term

Question

Given three equation

$$\log{(2xy)} = (\log{(x)})(\log{(y)})$$ $$\log{(yz)} = (\log{(y)})(\log{(z)})$$ $$\log{(2zx)} = (\log{(z)})(\log{(x)})$$

Find the real solution of (x, y, z)

What should I do to get the answer? and I think it's not possible that x = y = z has a solution, I have no idea what method I can do. Show me a hint

Answer 1

Note that $\log abc = \log a + \log b + \log c$

If you then substitute $\log x = a$, $\log y = b$, $\log z = c$, you get an equation in $a, b, c$ which is easier to solve.

As suggested, I would check the second equation-is a 2 missing?

Answer 2

Set $a=\log x$, $b=\log y$, $c=\log z$ then solve

• $\log 2+a+b=ab$

• $b+c=bc$

• $\log 2+a+c=ac$

subtracting the first and the third

• $b-c=a(b-c) \implies a=1 \lor b=c$

but $a=1$ is not acceptable then $b=c$.