The five essential laws: (1) Product: $log_a$(xy) = $log_a$(x) + $log_a$(y). (2) Quotient: log_a$$\frac{x}{y} = $log_a$(x) - $log_a$(y). (3) Power: $log_a$($x^n$) = n*$log_a$(x). (4) Change of base: $log_a$(b) = log_c$$\frac{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 2$log_a$(x) requires x > 0. This subtle domain difference is tested in JEE MCQs. All laws require positive arguments.