there is some confusion in the literature over the exact form defined for the lognormal distribution. First, it is necessary to pay attention to whether the distribution is in the natural logarithm or base 10 logarithm of the variable; the quoted value of $\sigma$ will be different in the two cases for the same distribution. Secondly, the way we have defined it in equation (\[eq:lognormal\]), the average luminosity is given by $L_0(M)$. This is due to the second term in the denominator of the exponential term, which not all authors include; if it is omitted, $L_0$ would not be the average luminosity.

The difficulty with using an approach such as this is that we are left with two unknowns we need to determine, the reference luminosity function $L_0$ and the dispersion $\sigma$. The only condition we can impose on this distribution is that, when integrated over all hosts, the resulting luminosity function must match the observed one. Fitting this calculated luminosity function to the observed one then allows us to relate the two parameters we have when we take $L_0=L(M)$ from equation (\[eq:mlmod\]), $a$ and $M_s$, to the scatter $\sigma$. However, this still leaves us with one free parameter which 