Predicting using an exponential model

I have the following data:

Days    Total cases1   32   33   54   65   286   307   318   349   3910  4811  6312  7013  8214  9115  10716  11217  12718  14619  17120  19821  25822  33423  40324  49725  57126  65727  73028  88329  102430  113931  132932  163533  205934  254535  310536  368437  428938  477839  535140  591641  672942  760043  845244  921045  1045346  1148447  1237048  1343149  1435350  1572451  1730452  1854353  2008054  21372

I defined days as 'days' and total cases as 'cases1'. I run the following code:

exp.mod <- lm(log(cases1)~days)

I get a good model with reasonable residuals and p-value.

but when i run the following:

predict(exp.mod, data.frame(days=60))

I get the value of 11.66476, which doesnt seem to be correct.

I need to get the value and also include the predictive plot in the exponential model.Hope that clarifies the issue.