The logarithm $log_a$(x) = y means $a^y$ = x. Three non-negotiable conditions: base a > 0, base a != 1, argument x > 0. The function y = $log_a$(x) is the inverse of y = $a^x$, so its graph is the reflection of the exponential across y = x. For base a > 1, the log function is increasing; for 0 < a < 1, it is decreasing. The domain (0, infinity) and range (-infinity, infinity) are fixed regardless of base. Key points on the graph: (1, 0) always lies on the curve, and (a, 1) is the reference point. In JEE, domain verification after solving log equations is the most critical skill — extraneous solutions that make arguments negative or zero must be rejected. The log function grows extremely slowly compared to polynomial functions.