Offhand, I don't know it either. But let me take a guess.

1. Start with a [log-normal distribution](https://en.wikipedia.org/wiki/Log-normal_distribution). It has the CDF
$$F(x;\mu, \sigma) = \frac{1}{2} \left[1 + \operatorname{erf} \left( \frac{\ln x - \mu}{\sigma \sqrt{2}} \right) \right]$$
2. Apply the [truncation formula](https://en.wikipedia.org/wiki/Truncated_distribution) on the interval $[a,b]$ $$F(x| a < X \leq b]) = \frac{F(x) - F(a)}{F(b) - F(a)}$$
which by composition gives

$$F(x;\mu, \sigma | a < x \leq b) = \frac{\frac{1}{2} \left[1 + \operatorname{erf} \left( \frac{\ln x - \mu}{\sigma \sqrt{2}} \right) \right] - \frac{1}{2} \left[1 + \operatorname{erf} \left( \frac{\ln a - \mu}{\sigma \sqrt{2}} \right) \right]}{\frac{1}{2} \left[1 + \operatorname{erf} \left( \frac{\ln b - \mu}{\sigma \sqrt{2}} \right) \right] - \frac{1}{2} \left[1 + \operatorname{erf} \left( \frac{\ln a - \mu}{\sigma \sqrt{2}} \right) \right]}$$

$$= \frac{ \left[1 + \operatorname{erf} \left( \frac{\ln x - \mu}{\sigma \sqrt{2}} \right) \right] -  \left[1 + \operatorname{erf} \left( \frac{\ln a - \mu}{\sigma \sqrt{2}} \right) \right]}{ \left[1 + \operatorname{erf} \left( \frac{\ln b - \mu}{\sigma \sqrt{2}} \right) \right] -  \left[1 + \operatorname{erf} \left( \frac{\ln a - \mu}{\sigma \sqrt{2}} \right) \right]}$$


$$= \frac{\operatorname{erf} \left( \frac{\ln x - \mu}{\sigma \sqrt{2}} \right)  -  \operatorname{erf} \left( \frac{\ln a - \mu}{\sigma \sqrt{2}} \right) }{ \operatorname{erf} \left( \frac{\ln b - \mu}{\sigma \sqrt{2}} \right) -  \operatorname{erf} \left( \frac{\ln a - \mu}{\sigma \sqrt{2}} \right) }$$