MathNet 2002 — Part 02 Question 86

Given that $a, b, c$ are positive real numbers and $\log_{a} b + \log_{b} c + \log_{c} a = 0$, find the value of $\left(\log_{a} b\right)^{3} + \left(\log_{b} c\right)^{3} + \left(\log_{c} a\right)^{3}$.