Select the cut element and apply X form modifier. This way u'll be able to animate the separate the element like an independent object.
But before animating...I might have told you move the key frame at zero to 1 or 2 (the keyframe for the axe), this way you'll be preventing key frames from coinciding (key frame for the cut animation & the X form). Turn on animate and go to frame 60, select the xform sub object. Now you can use move and rotate gizmos for freeform animating the element (like its falling on the ground). But for that be sure... make the animation start after frame 30... so move the initial key frame that is created at frame '0', to 31 or 32. Or you'll see abnormal mesh deformation... SO, this way you can get the required result.
Cutting a semi hard object.....like wax curving out while cutting (or carving)
1) Here we'll follow the same steps....but add a fews steps in between..
2) Ok..after you've animated the cut using BOOLEAN, here dont apply edit mesh modifier..instead you apply a volume select modifier...here you'll have the benefit that you dont have to rely on mesh count you can just select a volume of mesh..here use face selection type and use box type selection and scale it accordingly as explained in the picture. Place it just at the edge of the cutting.
3) There after you can apply a bend modifier on top and use its parameters and sub-gizmos to animate it curving out while cutting. But for that you've play a bit with the parameters to get the desired effect.
Cutting soft objects..(like cake) we'll again go back to the past and add something to it.. well this time you can model something like cake to cut off cutting, instead of a woddy cylinder. after you animated the axe or "knife" in the BOOLEAN sub-object then select the cake object from its sub-object. go down the modify stack to the editable mesh. There go to vertex mode and select the vertexes which might be affected while cutting the cake (ie. at the cutting edge). turn on "soft selection" increase it desirably and set the pitch to" .5". Now turn on animate and move to frame 30 and pull the vertices down..so as to simulate soft pressing effect. for separating the piece here u might slide the x-from gizmo instead of making it fall as a cake doesn't fall..while you cut it. The rest of the steps are the same.. and it's your creativity in how well you can cut a cake..