First, measure your fireplace opening and determine the size of plywood insert needed. If your fireplace already has a glass door front, you will want to measure the insert to fit exactly over this space. If your fireplace is open with brick or stone creating a lip, you may want to measure the insert to sit just behind this lip. Cut plywood to size, checking that the insert easily fits before moving on to the next step. See here for tips on cutting plywood.
This step will vary based on how your log slices are obtained. Whether you choose to cut salvaged logs, or buy craft discs (like these found online), you will want to end up with 50-100 pieces depending on your width and insert size. For the most realistic look, use a variety of log widths between 1"-6", and vary the depths between 1"-2". Keeping in mind you will want a nice, flat cut for the best glue adhesion, and the larger and thicker the slices are, the harder gravity will work against you. If you are using fresh logs from outdoors, you may need to let these slices dry out after cutting.
Once the paint is dry on your plywood, and you have gathered your wood slices, take some time to puzzle together a layout that looks natural. Start with spreading out the largest pieces and then fill in with the smaller ones, shuffling around until you have covered the plywood insert in a natural looking way. If you measured your insert to sit behind a lip, allow space on the edges of your plywood for this lip.
Once your layout looks good, one by one lift each slice and apply an ample amount of liquid nails or wood glue, pushing it down back into place until you have securely glued each slice. Use these tips on gluing wood if you are uncertain which adhesive to use, or how to prepare the surface correctly. Let dry flat a minimum of 24 hours.
When your wood glue is completely dry, and you have checked that all wood slices are firmly secure to the plywood, prop your insert up and install in your fireplace. If the logs you used are particularly heavy, you may want to add an additional finishing nail to each slice. If you live in a climate susceptible to high moisture or termites, you may want to finish the insert with a coat of clear polyurethane to seal the wood.