I have the same little log periodic, several actually and a similar dish. What you will need is something to hold the log antenna at the focal point of the dish that will allow you to move the log antenna in and out of the focal point. The phase center of a log periodic varies with frequency meaning it will have to be closer to the dish when using the lower frequency range of the log antenna and it will have to move away from the dish when using higher frequencies. You also have to find the focal point and peak the log antenna with the dish at all frequencies to know where the log antenna needs to be over its entire frequency range with some sort of marking so the adjustments are repeatable. The log antenna will also have to rotate to change polarity.
After the dish and log feed are married together you would need an LNA to cover all the frequencies of interest to drive a long cable back to your radio and preferably with a power inserter at each end to run power up the coax.
I also have a mil surplus 1.8m offset feed carbon fiber dish I'm saving for a project like this but in 20yrs I've not found the time to experiment finding the focal point and making a mount for the log periodic antenna. I know how to do it but its a daunting task that will probably take at least several days to figure out and I may never get to it.