Need some hints for proving a logarithmic inequality. $$\frac{\log_ax}{\log_{ab}x} +
\frac{\log_b{x}}{\log_{bc}x} +
\frac{\log_cx}{\log_{ac}x} \ge 6$$
Did as you suggested and got this, im stuck again:
$$\log_ab + \log_bc + \log_ca \ge 3$$

If you write $\frac{\log_a x}{\log_{ab} x}=\frac{\frac{\ln x}{\ln a}}{\frac{\ln x}{\ln ab}} = \frac{\ln ab}{\ln a} = 1+\frac{\ln b}{\ln a}$, and something similar with the other fractions, you can the use the familiar inequality :
$$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\ge3$$
which can be derived easily from the inequality between arithmetic and geometric means.
