rcollyer is right about the source of the problem, but there are better (faster) ways to handle it than merely cranking up the PlotPoints
. With that option alone I needed about PlotPoints -> 300
to get a smooth line, which took seven seconds to render:
eq = 2.` c^4 - 1.6931471805599454` c^4 x - 0.8611473146305157` x^2 +
0.041666666666666664` Log[10^(-6)] x^2 +
0.25` x^2 Log[1/c^(2/3)] + (1.` c^4 + 0.6732867951399863` x) x Log[x] -
0.125` x^2 Log[x]^2 == 0;
ContourPlot[Evaluate @ eq, {x, 3, 100}, {c, 1, 50},
PlotRange -> All, PlotPoints -> 300] //AbsoluteTiming //Column

MaxRecursion
helps a lot in this case:
ContourPlot[Evaluate @ eq, {x, 3, 100}, {c, 1, 50},
PlotRange -> All,
PlotPoints -> 75,
MaxRecursion -> 6
] // AbsoluteTiming // Column

Even better here appears to be controlling the lower level MaxBend
parameter:
ContourPlot[Evaluate @ eq, {x, 3, 100}, {c, 1, 50},
PlotRange -> All,
MaxRecursion -> 3,
Method -> {MaxBend -> 0.5}
] // AbsoluteTiming // Column
