First, let's look at your data. The Month column actually contains dates that are sometimes one day apart and sometimes a few months apart. It's not clear what that column should contain, but I don't think what you show is correct.
When you create the salesTS object you give it frequency=4 which would be appropriate for quarterly data, but the frequency should be 12 for monthly data.
The code to produce the forecasts should work ok.
Since you take logs before fitting the model, presumably you want forecasts back on the original scale, not on the log scale. You can easily do that by replacing the HoltWinters command with the equivalent model computed using the ets command. Here is an example. Note that lambda=0 is the same as taking logs.
library(ggplot2)
library(forecast)
sales <- data.frame(
Month = c(
"1982-06-12", "1983-06-19", "1983-06-20", "1983-06-21", "1983-06-22",
"1983-06-23", "1983-06-24", "1983-06-25", "1983-06-26", "1983-06-27",
"1983-06-28", "1983-06-29", "1982-09-04", "1982-09-05", "1982-09-06",
"1982-09-07", "1982-09-08", "1982-09-09", "1982-09-10", "1982-09-11",
"1982-09-12", "1982-09-13", "1982-09-14", "1983-11-20", "1983-11-21",
"1983-11-22", "1983-11-23", "1983-11-24", "1983-11-25", "1983-11-26",
"1983-11-27", "1983-11-28", "1983-11-29", "1983-11-30", "1983-12-01",
"1983-12-02", "1983-12-03", "1983-12-04", "1983-12-05"
),
sales = c(1L, 6L, 2L, 4L, 3L, 5L, 8L, 9L, 4L, 5L, 6L, 1L, 5L, 3L, 4L, 6L,
5L, 6L, 4L, 1L, 2L, 2L, 2L, 1L, 3L, 4L, 4L, 7L, 10L, 10L, 9L, 7L,
8L, 2L, 3L, 4L, 1L, 1L, 1L)
)
salesTS <- ts(sales$sales, frequency = 12, start = c(1982,1))
salesLogHW <- ets(salesTS, model="AAA", lambda=0)
nextYearSales <- forecast(salesLogHW, h=4)
autoplot(nextYearSales)

nextYearSales
#> Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
#> Apr 1985 5.103658 1.866121 13.95800 1.095562 23.77531
#> May 1985 5.554407 2.030935 15.19076 1.192320 25.87513
#> Jun 1985 4.704604 1.720209 12.86663 1.009900 21.91633
#> Jul 1985 5.758174 2.105440 15.74804 1.236061 26.82437
Created on 2020-04-20 by the reprex package (v0.3.0)