Log files location can be easily specified in Central Administration: Monitoring – Configure Diagnostic Logging.
In the Diagnostic Logging screen, you can specify a new path for trace log files and change other settings, such as how long to keep trace files, and what sort of information will be logged there.
Index files location, unfortunately, is not available through Central Administration. It is typically specified during SharePoint installation phase, along with the software installation folder. However, there is a way to change it through stsadm command:
stsadm -o spsearch -indexlocation “d:\SharePoint\Data”
You can verify that it changed successfully by running the following command before and after changing the index:
I wanted to thank Tracy Sterling for submitting a very useful tip for modifying index locations through Central Administration, when you have farm SharePoint deployments and can alter search topology.
CA > Application Management > Manage service applications > Search Service Application > Search Application Topology (Modify) > Index Partition/Query Component (Edit Properties). Repeat for each query component within your farm.
Unfortunately, all the stand-alone deployments and foundation editions would still have to go through aforementioned stsadm command. Below is an illustration of Search Service Application and its missing modify option for a stand-alone installation of enterprise SP2010 server, running on my laptop.