log_a(xy) = log_a(x) + log_a(y), log_a(x/y) = log_a(x) - log_a(y), log_a(x^n) = nlog_a(x). These hold only when all arguments are positive. Common error: writing log(x^2) = 2log(x) without noting that log(x^2) is defined for all x != 0, but 2*log(x) requires x > 0. In JEE, this distinction matters when the original expression has x^2 inside the log.