Option 1 states that an antibiotic is a chemical compound produced by a living organism that inhibits the growth of other organisms. This is generally correct, as antibiotics are indeed produced by living organisms and are designed to inhibit the growth of other organisms, particularly bacteria. However, it is important to note that antibiotics specifically target bacteria and not all organisms in general.

Option 2 states that an antibiotic is a compound synthesized by a living organism that inhibits the growth of microbes. This option is also correct, as antibiotics are indeed synthesized by living organisms and are effective against a variety of microbes, including bacteria, fungi, and some parasites.

Option 3 states that an antibiotic is a synthetic compound inhibiting the growth of other organisms. This option is incorrect, as antibiotics are not exclusively synthetic compounds. They can be either naturally produced by living organisms or synthesized artificially.

Therefore, option 4 is the correct answer, as it accurately states that antibiotics are synthetic compounds that specifically target and inhibit the growth of bacteria.