The forest service rangers do not wish folks to leave the trail to do earthcaches. You can make your observations from the trail.
The geologic story of the Red River Gorge area began over 300 million years ago during the Carboniferous Period of the Paleozoic Era. Paleozoic sediments that eroded from the ancestral Appalachian Mountains located to north and east in Pennsylvania were transported by rivers flowing south into eastern Kentucky where these rivers emptied into a shallow sea. Here, during a portion of the Carboniferous Period called the Pennsylvanian Age, a river delta plain was formed. Layer upon layer of quartz sand and gravel sediments built up in this delta plain and eventually formed a thick deposit. Over time this thick deposit of quartz sand and gravel became consolidated and cemented into a quartzose sandstone that has been designated by geologists as the Corbin Sandstone Member of the Lee Formation. But the formation of the Corbin Sandstone is just the beginning of the Red River Gorge geologic story. The entire Eastern Kentucky area was uplifted to form the Cumberland Plateau. This uplifting resulted in an extensive vertical fracturing of the Corbin Sandstone. As the uplifted plateau was subjected to weathering and erosion, clifflines formed along the vertical fractures. In many places the crisscross pattern of these fractures results in a saw tooth cliff outline. In several places this pattern of fracturing has also resulted in free standing pinnacles.
The cave you are standing before is the side of one of these pinnacles. The "cave" or rock shelter was formed in much the same way as the vertical features.
Over time, water slowly eroded the sandstone forming a cliff or bluff above the valley (Stage 1). The upper sandstone stratum was more resistant than the lower stratum which lies just below. Thus, the lower stratum more quickly eroded allowing the upper stratum to be undercut (Stages 2-5). This undercut forms the rockshelter. Weathering will continually undercut the less resistant stratum while also eroding the other resistant strata. Eventually portions of the roof will begin to collapse and fall to litter the floor of the rockshelter (Stages 6-8). The final phase or the death of a rockshelter will be when the roof has a total collapse and there is no longer an alcove present (Stage 9).

To log a find email the answers to these questions.
1) Based on the diagram above - what stage of rock shelter would you estimate this rock shelter to be?
2) From the front lip of the roof of the shelter to the floor - how high would you estimate the opening of the over hang?
Getting to these trails is deceiving based on local maps. At first glance, you would think that you could make it to these trails off of 77, however a one lane tunnel passes under the trail entrance. To get there, you need to follow take 15 off of the Mountain Parkway and follow it down to Tunnel Ridge Road
DIRECTIONS TO TRAILHEAD: From rest area at exit #33 off of the Mountain State Parkway Turn left, and go to end of road (.3 miles) Turn Right onto KY 15 (3.3 miles) Turn Left onto Tunnel Ridge Road (3.7 miles) Park at the End of Tunnel Ridge Road in the Auxier Ridge Parking Area
For up to date information either stop in the Ranger District Office, visit their website or telephone: (859) 745-3100