Simplifying logarithmic equations

Question

$$(\log x) ^2 (\log y + \log z) =1$$

$$(\log y) ^2 (\log x + \log z) =2$$

$$(\log z) ^2 (\log y + \log x) =5$$

Find $\log x \cdot \log y \cdot \log z$ ?

It is fairly obvious that the problem could have several answers.

A difficult method would be to solve three equations in three variables and calculating the product.

But how can I make use of certain logarithm properties to ease the search of the solution?

Answer 1

Let $\log{x}=a$, $\log{y}=b$ and $\log{z}=c$.

Hence, $a^2b^2c^2(a+b)(a+c)(b+c)=10$ and $\sum\limits_{cyc}(a^2b+a^2c)=8$.

Thus, $a^2b^2c^2(8+2abc)=10$, which gives $$(abc-1)(a^2b^2c^2+5abc+5)=0$$

Answer 2

Setting $$\log(x)=a,\log(y)=b,\log(z)=c$$ then we have to solve

\begin{align}a^2(b+c)&=1\\ b^2(c+a)&=2\\ c^2(a+b)&=5\end{align}

Eliminating $b,c$ from the system then we get, for $a$

$$13a^3-101a^6+9a^9+2=0$$

Can you solve this?