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 u move the key frame at zero to 1 or 2(the keyframe for the axe) , this way u'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 u 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 u'll see abnormal mesh deformation...SO, this way u can get the required result.
see the video file for visual demonstration.

PART 2

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 u've animated the cut using BOOLEAN, here dont apply edit mesh modifier..instead u apply a volume select modifier...here u'll have the benefit that u dont have to rely on mesh count u 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 u can apply a bend modifier on top and use its parameters and sub-gizmos to animate it curving out while cutting.but for that u've play a bit with the parameters to get the desired effect.

see the video file for a visual idea.

PART 3

cutting soft objects..( like cake)

we'll again go back to the past and add some thing to it..well this time u can modell some thing like cake to cut off cutting, instead of a woddy cylinder. after u 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 u cut it. rest steps r the same..and its your creativity in how well u can cut a cake..
see the video file for visual demonstration. in the video i've textured the cake. see the texturing part to learn that.