I wrote an article about the ability to enable Web Server Logging, Detailed Error Message, Failed Request Tracing and Application Logging here. In that article I discussed that you can download the logs using an FTP tool like FIlezilla. You can also download the log files from within Visual Studio by performing the following. I currently have the Windows Azure SDK 2.1 installed, expand the Windows Azure Web Sites menu item within Server Explorer, then expand the subscription and right-click on the web site you want to down load the logs for. Lastly, click on the View Settings… menu item shown in Figure 1.
Figure 1, View Windows Azure Web Site details in Visual Studio
By default this opens a window showing the current configurations applied to the selected web site. Figure 2 illustrates this.
Figure 2, Windows Azure Web Sites configuration page in Visual Studio
Notice that there is also a Logs item feature available for use. Clicking the item renders the page similar to that shown in Figure 3.
Figure 3, Download Logs from a Windows Azure Web Site using Visual Studio
Notice that there are 3 available options.
There is also an option to enable the Diagnostics on the Web Site. This supports the viewing of events logged into the event viewer or custom events. This feature requires the linking of a storage resource to the web site and the configuration of the Site Diagnostics to be configured to store the log into a Blog container. An article on this is in the works.