Convert the dates to duration from the starting date but use the numeric form of the durations (ie, not the "duration' class). Then you're just working with regular old numbers that can sustain the log scale.
If you want to see the date for each point, add the dates to the datatips.
dt = datetime(2020,1,1) + days(0:5:365);
data = exp(1:numel(dt));
ndays = days(dt - dt(1));
h = loglog(ndays,data, 'o');
xlabel(sprintf('Number of days since %s',datestr(dt(1))))
h.DataTipTemplate.DataTipRows(end+1) = dataTipTextRow('Date',dt);
X is the number of days since the first datetime value.
Date shows the datatime value for each data point.