Just to understand: If a living connection goes down, wouldn't it be easier to log that on the client side?

I don't know which API you are using. AFAIK, v11 has introduced a ODBC feature that might be of use in case you are using ODBC - from the "What's new / Programming interfaces" doc page:

SQL_ATTR_CONNECTION_DEAD promptly detects dead connection

Using ODBC's SQLGetConnectAttr call to get the SQL_ATTR_CONNECTION_DEAD attribute now gets the value SQL_CD_TRUE if the connection has been dropped even if no request has been made to the server since the connection was dropped. Determining if the connection has been dropped is done without making a request to the server, and the dropped connection is detected within a few seconds. The connection can be dropped for several reasons, for example, on an idle timeout. Before this change, SQL_ATTR_CONNECTION_DEAD only got the value SQL_CD_TRUE if the connection was disconnected or if ODBC driver made a request to the server (by calling SQLExecDirect for example) after the connection was dropped. See Getting connection attributes.

If the network goes down between our application host and database host briefly I would like to see an appropriate error in the database logs.

I believe you really mean "application logs" here, not database server logs. Again, the failure to contact the server over TCP/IP from the client needs to be logged on the client side - the database server can't log a connection failure that it can't see.

Beyond the "LOG=" client log connection string option that I mentioned previously, depending on which API you're using (as Volker indicated), you can usually 'catch' some kind of exception / log an error upon a failed connection attempt from the client side in the API layer, and then "log" it somewhere in your application.

e.g. ADO.NET C# code:

SAConnection conn = new SAConnection( "Data Source=SQL Anywhere 12 Demo;UID=DBA;PWD=sql" );

try {
    conn.Open();
    ...
} catch (SAException ex){
    MessageBox.Show( ex.Errors[0].Message );
    // do more application log error logging here
}

e.g JDBC code:

try {
    Connection conn = DriverManager.getConnection( "jdbc:sqlanywhere:uid=DBA;pwd=sql" );
    ...
} catch (SQLException sqe){
   System.err.println("Error: " + sqe.toString() + ", sqlstate = " + sqe.getSQLState());
    // do more application log error logging here
}