Question

A Log floats in a river with one-forth of its volume about the water. (a) what is the density of the log?

Answer 1

if one-fourth of the volume is above water then I think the density is 25% less than that of water
I never took any fluid mechanics though, so I'm not sure...
you say about though, so I'm not sure the meaning of your question

Answer 2

So summing up the forces,
\[ mg = \rho_w V_s g\]
where rho_w is the density of water and V_s is the volume of the log that's submerged. Rewriting the left side,
\[\rho_lV_t g = \rho_wV_sg\]
where rho_l is the density of the log and V_t is the total volume of the log. Cancelling g and manipulating, we find that
\[\frac{\rho_l}{\rho_w} = \frac{V_s}{V_t} = \frac{3}{4} \]
so
\[\rho_l = \frac{3}{4} \rho_w \]
and indeed, as Turing said, the density of the log is 3/4 the density of the water.

Answer 3

This is of course neglecting things like surface tension which are very important but I suspect are not being taken into account here :)