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I built a log home about a decade ago but, unfortunately was not around for the wiring stage.
Termites might do the trick but are not really predictable enough.
You didn't say what the chinking space was between the logs??
I would think that the box would usually st in the chinked portion of the structure.
If this is not desireable then simply drill on, as high an angle as possible, down through the box cavity using a large forstner bit and then eye-ball an upward diagonal hole from the chink space below the box. If the bit is big enough you shouln't have too much trouble aligning them enough to get your wire through.
Maybe an even simpler method would be to place the box so that either the top or the bottom lands at, or near the edge of the log.
It just dawned on me that you may be using round logs. In this event ignore the final idea