In order to send external email you will need to configure a 'Send Connector'.
In my case I need to relay my outgoing email through my ISP's email server so that when my mail reaches its destination the reverse DNS Look-up frequantly used by spam filters does not classify my mail as spam because it originated from a broadband IP Address.
To Configure a 'Send Connector'
Open Exchange Admin Center and select 'Mail Flow' from the left menu and then 'send connectors' from the top menu.
Click the + to make a new send connector.
Set the name you wish to give the send connector and select 'Internet (For example, to send internet mail)' and click next.
Select '
Click the + to add the address of the smart host you wish to relay through.
Type the name of the destination mail server for instance mail.yourisp.net and click save.
Click 'next'
In my case to authenticate to the destination server I need to use 'Basic Authentication' so select your method and enter your 'Username' and 'Password'
Click 'next'
In order to tell the connector what domains you want to use this connector for you need to specify the 'address space' because its the internet it will need to send anything so click + and in the '
*Full Qualified Domain Name (FQDN):' there a '*' (without quotes)
Click 'save' then 'next'
In the 'source server' box you will need to add all of your servers. So click the + and add them in.
Click finish
Now you have a send connector set-up your servers will be able to send messages from your internal network to the internet.