I will consider this one valid -albeit a quite old unanswered question- because I had the same issue. Installed Linux Mint and didn’t find Logseq in Package Manager nor with APT so I ended up searching for a solution.
I ended up on github, where evdcush/logseq_install_update.sh has posted a script to download and install Logseq directly from Github/Logseq. The script installs or updates Logseq so it will be handy in future too.
The only thing I had to troubleshoot is at line 158:
convert -resize 512x512 "${TEMP_DIR}/logseq.png" "${TEMP_DIR}/logseq.png"
As Mint doesn’t have, by default, imagemagik, i didn’t have convert so i ended up using ffmpeg, which exists on a vanilla Mint, so comment the abov or replace with the below:
#convert -resize 512x512 "${TEMP_DIR}/logseq.png" "${TEMP_DIR}/logseq.png"
ffmpeg -loglevel error -i "${TEMP_DIR}/logseq.png" -vf scale=512:512 "${TEMP_DIR}/logseq_tmp.png" && mv "${TEMP_DIR}/logseq_tmp.png" "${TEMP_DIR}/logseq.png"
NOTE: ffmpeg can’t convert in place so a two-step was necessary.
Hope it helps someone 