It is possible to use a DB connection string specified in web.config without creating a new class, though you would need to use log4net build that hasn’t been released yet. It can be downloaded from SVN repository http://svn.apache.org/viewvc/logging/log4net/trunk/
Your config will look as follows:
<connectionStrings>
<add name="connStr" connectionString="Data Source=localhost; ..." />
</connectionStrings>
<log4net>
<appender name="AdoNetAppender" type="log4net.Appender.AdoNetAppender">
<connectionType value="System.Data.SqlClient.SqlConnection, System.Data, Version=2.0.0.0, Culture=neutral, PublicKeyToken=b77a5c561934e089" />
<connectionStringName value="connStr" />
...
</log4net>
Please note that connectionType still needs to be specified.