The modified lognormal power-law (MLP) function is a three parameter function that can be used to model data that have characteristics of a log-normal distribution and a power law behavior. It has been used to model the functional form of the initial mass function (IMF). Unlike the other functional forms of the IMF, the MLP is a single function with no joining conditions.

The closed form of the probability density function of the MLP is as follows:

where ${\displaystyle {\begin{aligned}\alpha ={\frac {\delta }{\gamma }}\end{aligned}}}$ is the asymptotic power-law index of the distribution. Here ${\displaystyle \mu _{0}}$ and ${\displaystyle \sigma _{0}^{2}}$ are the mean and variance, respectively, of an underlying lognormal distribution from which the MLP is derived.

Following are the few mathematical properties of the MLP distribution:

The MLP cumulative distribution function (${\displaystyle F(m)=\int _{-\infty }^{m}f(t)\,dt}$) is given by:

We can see that as ${\displaystyle m\to 0,}$ that ${\displaystyle \textstyle F(m)\to {\frac {1}{2}}\operatorname {erfc} \left(-{\frac {\ln(m-\mu _{0})}{{\sqrt {2}}\sigma _{0}}}\right),}$ which is the cumulative distribution function for a lognormal distribution with parameters μ₀ and σ₀.

The expectation value of ${\displaystyle M}$^k gives the ${\displaystyle k}$^th raw moment of ${\displaystyle M}$,

This exists if and only if α > ${\displaystyle k}$, in which case it becomes: