You are calculating log(0) in your second example. The code will work in some Plots.jl backends (ex: pyplot() or pgfplotsx()) if you take care of the zeros.
One option is to use scale=:log10 in your first example and to exclude the zeros using appropriate xlims and ylims.
Example using Plots; pyplot():
scatter(means_star', variances', label="", xlabel="μ*", ylabel="σ", series_annotations=text.(pn,:top,12), scale=:log10, xlims=(1e-10, 1e10), ylims=(1e-20, 1e20))
