Irrational is the value of לג (the Biblical unit). Allow me to explain:
We find five verses with logs in the Torah (Lev 14:10, 12, 15, 21, and 24), and they are all logs of oil. So if we let x be the value of לג, then x = log (oil).
To figure out what x is, we have to determine the value of oil. Oil, when not otherwise specified, must mean anointing oil (שמן המשחה), which was comprised of a total of 1500 shekels and 1 hin (Ex 30:23-24). Now, 1 hin is equal to 12 logs, which means that oil has the value of 1500 + 12x (by an abuse of units).
One final item before we solve for x: with what base do we use the logarithm? Well, in Lev 14:10, we find "ולג אחד שמן". Since לג is already in singular, do we not already know that one is meant? Rather, the אחד must come to tell us about the base (note also its placement, right after לג). Since log base 1 is meaningless, the natural logarithm must be intended, ie. base e.
Taking the exponential of both side of x = log (1500 + 12x) leads us to the equation e^x = 1500 + 12x. Whenever x is rational, the right-hand side is rational, whereas the left-hand side is irrational. Thus, we conclude that x is irrational.