Q:

How solve this logarithms equation

What relationship between a,b and c ?

A:

From Lucian's suggestion to transform $\log_a u=\frac{\ln u}{\ln a}$ etc. we find, using $x=\ln a, y=\ln b, z=\ln c$ and cancelling the common $\log u$ factor, that
$$\frac1x+\frac1y+\frac1z-\frac{1}{x+y+z}=0.\tag{1}$$
The left side factors into
$$\frac{(x+y)(x+z)(y+z)}{xyz(x+y+z)}.\tag{2}$$
Now since we are working with $a,b,c$ being the base of logs, we are assuming each is positive and not $1$, so that $x,y,z$ are nonzero. We also know $x+y+z \neq 0$ since it is the log of $abc$ which also appears as a logarithm base in the original equation.
So the top of $(2)$ must be zero, which means some two of the three logs add to zero, which in turn means that at least one of the three possibilities $ab=1,ac=1,bc=1$ must hold. It is then easy to check that on the other hand provided one of the three products is $1$ then the statement holds for all $u$.

