In this case, ContourPlot can do the trick.

     f[z_] := Log[z^3];
     With[{z = x + I y}, ContourPlot[Im[f[z]], {x, -2, 2}, {y, -2, 2}, ClippingStyle -> Automatic, ExclusionsStyle -> {Black, Dashed}, 
     Axes -> True, Frame -> False, Contours -> 1, ContourShading -> None,ContourStyle -> None, AxesLabel -> {"Re[z]", "Im[z]"}]]

Package ComplexAnalysis does not appear in the list of Standard Extra Packages, no file ComplexAnalysis.m seems to be in $InstallationDirectory, and the Documentation Center includes no entry for either ComplexAnalysis or BranchCuts!

However, if one evaluates an expression using function BranchCuts in context ComplexAnalysis, then

 ?ComplexAnalysis`BranchCuts

does reveal a usage message.

Very strange!

Very cool! Where/how did you find the function?

thank you so much Mr Huisman.

I am aware of the arbitrary way branch cuts can be defined.

Commonly though the Log[Z] function is defined on the principal branch with the branch cut being the origin and the negative real axis.

I used your code and graphed Log[Z] and it appears the branch is at y=0 and x<=1.

Why is this?

I thought it should be at y=0 and x<=0.

thanks in advance best wishes

i do not wish for a Riemann surface representation...just a 2d plot of the branch cuts like i have seen in Maple