Here are the two logs. Thank you again for your help.
How is the computer behaving now ?
The computer appears to be booting up and logging on faster. It had slowed when bringing up different websites and just running slower.
Would you please recommend software that we should run on the computer to keep it clean and run daily besides Avast. Right now DH only runs CCleaner. I know that is not doing the job. Other than the Avast warnings I/we did not know the computer had been infected. Thank you for all your help. So the last two files I uploaded looked pretty clean? I am glad. Thank you for your help. It was fantastic.
What I will do now is remove my tools. Once done you can set IE to empty browsing data on exit. I will post that at the end
Subject to no further problems 
I will remove my tools now and give some recommendations, but, I would like you to run for 24 hours or so and come back if you have any problems
Now the best part of the day ----- Your log now appears clean 
A good workman always cleans up after himself so…The following will implement some cleanup procedures as well as reset System Restore points:
Download and run Delfix
https://dl.dropboxusercontent.com/u/73555776/delfix.JPG
: Keep Java Updated :
WARNING: Java is the #1 exploited program at this time. The Department of Homeland Security recommends that computer users disable Java
See this article and this article.
I would recommend that you completely uninstall Java unless you need it to run an important software.
In that instance I would recommend that you disable Java in your browsers until you need it for that software and then enable it. (See How to diasble Java in your web browser and How to unplug Java from the browser)
Now that you are clean, to help protect your computer in the future I recommend that you get the following free programmes:
CryptoPrevent install this programme to lock down and prevent crypto ransome ware
https://dl.dropboxusercontent.com/u/73555776/CryptoPrevent.JPG
Update and run weekly to keep your system clean
It is critical to have both a firewall and anti virus to protect your system and to keep them updated.
To learn more about how to protect yourself while on the internet read this little guide Best security practices Keep safe 
Set IE to clear data :
Go to Control Panel > Internet Options > Select Advanced Tab
Scroll down to Security and tick Empty Temporary Internet files when closing
OK out
Then every time IE is closed all the temporary internet stuff will be gone
I will download Delfix. We do not use Internet Explorer. We predominately use Google Chrome and FireFox. It was the Firefox that was giving us the problem.