Aplicate a complex operation on a column to create a new dataframe in R

Question

I have a dataframe that looks like the the following, lets call it DF

Symbol	Date	volume	price
A	2014-01-01	1	5
A	2014-01-02	3	8
A	2014-01-03	7	4
A	2014-01-07	3	6
A	2014-01-08	34	7
A	2014-01-09	45	34
A	2014-01-10	4	5
A	2014-01-11	9	7
A	2014-01-14	8	6
A	2014-01-15	4	4
A	2014-01-16	0	7
A	2014-01-17	4	7

I want to apply the next operation to the price column:

diff(log(DF$price))

And generate a datafra like this:

Date	returns
2014-01-01	4
2014-01-02	6
2014-01-03	8
2014-01-07	2
2014-01-08	14
2014-01-09	5
2014-01-10	1
2014-01-11	2
2014-01-14	8
2014-01-15	4
2014-01-16	0
2014-01-17	4

(The numbers, in the column returns i the examples not are the desired one, are just random numbers that i put to ilustrate, i need that in each row of that column appears the correspondent result of diff(log(DF$price)))

I tried the next lines of code but it did not result:

Ret <- DF %>% group_by(date) %>%
  mutate(Ret_i = diff(log(DF$price)))%>%
  summarise(Ret_i)

Thanks for reading and I would apreciate any help