LetA 1 andA 2 be floating point numbers represented in arbitrary base β and randomly chosen from a logarithmic distribution. Letr denote the round-off error $$r = fl(A_1 * A_2 ) - (A_1 * A_2 )$$ where * is floating point multiplication and wherefl(A 1*A 2) denotes the normalizedN digit computer result of forming (A 1*A 2). This paper analyzes the mean and variance of both the actual round-off error and the fraction round-off error. This analysis relies upon sharp order estimates for the digit by digit deviation of logarithmically distributed numbers from uniformly distributed numbers. This completely resolves open questions of Kaneko and Liu and of Tsao. Also included is a generalization to arbitrary base (from binary) of an important round-off theorem of Henrici.