Input in the top-box is the probability of expecting zero events, a number between zero and one, the value in the Expected box can be any (decimal) number larger than one representing the expected mean or average. The expected can also be entered as the distribution parameter, a value between zero and one. Input for the number Observed must be an integer value, a positive number without decimals. If you check always mean the expected is presumed always to be a mean.
The logarithmic distribution presented here is the zero modified version, the usual logarithmic distribution has a probability for the value zero of zero. The value one is the mode of the proper logarithmic distribution and thus has the highest probability value in the output, any subsequent value has a lower probability value as the previous value up to infinity. The logarithmic distribution is an ordinal version of the continuous power law distributions, such as the exponential and the pareto distribution. It gives you the probability of the elements in an ordinal ranking, from commonest to rarest object. There are a number of applications documented in the literature one of which is to model the dominance of species in biology. The probability of value one then stands for the probability that you catch the most dominant species, the probability of value two for catching the second most dominant species, three for the third most dominant species, etc. etc. The higher the distribution parameter or mean the flatter the distribution, the higher the probability that you catch some rarer species and the greater the evenness in the system. Note that the logarithmic distribution presumes that the number of different species is in fact infinite, there is no rarest species, there are always more rare species. Besides the logarithmic distribution for modeling the dominance of species there are competing models which are said to do the same in a different way and which can also be used to model ordinal power law functions, such as the geometric series, the broken-stick, or the log-normal model.
Unfortunately, the distribution parameter and the distribution mean have no easy empirical meaning. The expected mean for the proper logarithmic distribution is always larger than one, however, for the zero modified version it can be smaller than one if the probability of p0 is relatively high.
The logarithmic distribution is also known as the logarithmic series distribution or the log-series distribution.