You can’t send email to external address if you don’t add send connector to Exchange Server 2010. I will show you how you can easily add send connector with Powershell:

New-SendConnector –Name ‘External’ –Usage ‘Internet’ –AddressSpaces ‘SMTP:*;1’
–DNSRoutingEnabled $true –UseExternalDNSServersEnabled $false –Fqdn ‘mail.yusufozturk.info’

You can modify name and fqdn. My FQDN is the external domain name of my Exchange Server 2010.