Apple's iTunes software allows you to create iPhone backups on your computer, but computers have no native support for reading these encrypted backups. This means that you cannot view backed-up data, such as sent messages, unless you use a third-party tool. Apps like iMessage, Dr. Fone for iOS, iExplorer, DiskAid and CopyTrans allow you to view logs of your sent messages on your computer.

IMessage

If your iPhone and Mac computer are connected to an iMessage account, then you can check your sent message history by launching the Messages app on your computer. This native OS X application contains conversation logs from all iMessages sent between your devices and your recipients' Apple devices.

Dr. Fone for iOS

Download and install the free trial of the Dr. Fone software from the Wondershare developer website (link in Resources). The trial version allows you to view sent text messages, but you can only save these logs to your computer with the premium version, which costs $69.95 as of December 2013. Launch the application and plug your iPhone into the computer with your USB cable. Click "Start Scan" and click on Messages. Logs of your sent and received messages appear on the screen.

IExplorer

Purchase, download and install the iExplorer software from the developer website (link in Resources). The app costs $34 as of December 2013. Connect your iPhone to the computer with the USB cable and launch iExplorer. Choose the "Data" tab and click on "Messages." The history of your sent and received messages appears onscreen. You can select these messages individually or in conversation batches and save them to your computer as PDF or TXT files.

DiskAid

Purchase and download the DiskAid utility from the developer website, which costs $30 as of December 2013. Install the software and connect your iPhone to your computer with the USB cable. Launch DiskAid and click the "Messages" menu option, which appears under the "Data" sidebar category. This displays a history of your sent and received text messages.

CopyTrans Contacts

Purchase and download the CopyTrans Contacts application from the developer website (link in Resources). This software costs $15 as of December 2013. Install the software and launch CopyTrans Contacts. Connect your iPhone to your computer with the USB cable and click on the messages icon, which looks like a comic speech bubble. Your text message history appears onscreen, including all of your sent and received messages.