You don't need to memorize the properties of logarithms. Just change the numbers yourself and use a calculator to check whether both sides always come out equal, and you'll get a feel for why these properties hold.
log₂8=3 means "multiplying 2 by itself 3 times gives 8." A logarithm is really a calculation that asks "what power do I need to raise this to, to get this number?" But look at log₂8=3, log₂16=4, and log₂128=7, and you'll notice 3+4=7. That's not a coincidence — since 8×16=128, this shows log(M×N) = logM + logN: multiplication turns into addition.
The properties that turn division into subtraction, and a power into multiplication, come from this exact same principle. Before computers existed, this property let people turn hard multiplication into easy addition, and even today, logarithms are used whenever numbers span an enormous range — like the magnitude of an earthquake or the loudness of a sound. Change the numbers with the sliders and use the calculator to check for yourself whether both sides really always come out equal.