f(xy) = f(x) + f(y) for all x,y > 0. This is the defining property of the logarithm. Setting x = y = 1: f(1) = 0. Setting y = 1/x: f$\frac{1}{x}$ = -f(x). With continuity, f(x) = k*ln(x) for some constant k. If f(a) = 1 for some a, then f(x) = $log_a$(x). JEE problems typically give f at one or two points to determine the function.