model=D[n[t],t]==r n[t] (1-n[t]/k)

solution=DSolve[{model,n[0]==n0},n[t],t]

equation=n[t] /. solution[[1,1]]

times={25,30,40,50}

volumes={20,100,1000,900}

logvolumes=Log10[volumes]

data=Transpose[{times,logvolumes}]

p1=ListPlot[data,PlotMarkers->{Automatic,Medium},AxesOrigin->{0,0}]

fit=FindFit[data,{Log10[equation], n0>0,r>0,k>0},{{n0,0.01},{r,0.1},{k,1000}},t]

p2=Plot[Log10[equation] /. fit,{t,0,50}]

Show[p1,p2]