How can I draw a plot with Logarithmic scales in Juliabox?

You are trying to plot the value 0 (the first element in w_wn) on a logarithmic scale, which is impossible since it would have to be drawn at negative infinity. You get the same error message from

plot( [0], [1], xscale=:log10 )

Also, you probably don’t need to call collect on your range.
If you want to initialize beta1 as the same size as w_wn in order to fill it in a loop you can use similar(w_wn), or zeros(size(w_wn)...). But you can also initialize beta1 directly with a broadcasting computation on w_wn instead of that loop.