The five essential laws: (1) Product: log_a(xy) = log_a(x) + log_a(y). (2) Quotient: log_a(x/y) = log_a(x) - log_a(y). (3) Power: log_a(x^n) = n*log_a(x). (4) Change of base: log_a(b) = log_c(b)/log_c(a). (5) Reciprocal: log_a(b) = 1/log_b(a). The chain rule log_a(b)log_b(c) = log_a(c) collapses telescoping products of logarithms. Warning: log_a(x+y) != log_a(x) + log_a(y) — there is NO addition rule. Also, log_a(x^2) has domain x != 0, but 2log_a(x) requires x > 0. This subtle domain difference is tested in JEE MCQs. All laws require positive arguments.