Solve the following system

Question

Solve the following system in $$x, y, z$$ : $$2 ^ x 3 ^ y 5 ^ z = 2 ^ y 3 ^ z 5 ^ x = 2 ^ z 3 ^ x 5 ^ y = 30$$

I have tried to solve it but it seems impossible to solve.

Answer 1

If $x,y,z$ are integers, then it is easy, using the prime decomposition of $30$, to see that $x=y=z=1$ is the only solution.

If $x,y,z$ are not integers, take the $\ln$ : you get a system of three equations with three unknown, which has a unique solution. You can guess what it is :)

Answer 2

This is three equations wrapped into one. Taking the log of both sides gives

$$x\ln 2+y\ln 3+z\ln 5=\ln 30$$ $$x\ln 5+y\ln 2+z\ln 3=\ln 30$$ $$x\ln 3+y\ln 5+z\ln 2=\ln 30$$

This has a matrix equation of

$$\begin{pmatrix} \ln 2 & \ln 3 & \ln 5 \\ \ln 5 & \ln 2 & \ln 3 \\ \ln 3 & \ln 5 & \ln 2 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \\ \end{pmatrix}=\begin{pmatrix} \ln 30 \\ \ln 30 \\ \ln 30 \end{pmatrix}$$

To guarantee a solution we require that the determinant of the matrix is nonzero, and its determinant is

$$D=\ln^32+\ln^33+\ln^35-3\ln2\ln3\ln5\approx2.15\neq 0$$

Therefore there is a unique solution to the problem. We can guess the answer is $x=y=z=1$, or we can solve the system using row reductions, which will be quite annoying but systematic.