1. Obtain rough estimates of the mean and variance using the sample by any reasonable method.
2. Now write a Monte-Carlo routine that given the mean and variance of a log-normal generates a large sample of samples of size 10 and calculate the (same) rough estimates of the mean and variance for each sample of size 10, which when averaged give us an estimate of the biases and so a new estimate of the mean and variance. Now loop around the MC routine (starting with our initrial rough extimates) adjusting until the estimates of the mean and variance are not changing much..
The resulting estimates (of the mean and variance) will be (approximately) unbiased
Handwaving explanation: What we are doing is assuming some guess at mean and variance for the log-noormal and any old estimate using Monte-Carlo to estimate the bias, using this to correct the bias in our estimate to give us our new guess at the mean and variance of the log-normal and repeating until we have acceptable convergence (convergence will never be perfect because of the MC element of this process