According to 9to5Google, the new version of Duo (14) integrates with your Android phone's standard call log and will keep track of your Duo video calls once you give the app permission to do so. You'll see your Duo calls in your call log history, right next to the rest of the ones you make with your phone app. The video calls will have a cute little video camera icon under the number you called with Duo to help differentiate them from regular calls. The update is available from the Google Play Store as well as from APKMirror, a third-party mirror for updates like this.
Update: A Google spokesperson got back to us to confirm that Duo will add your video and audio calls to your regular phone log for Android and iPhone. A quick check this morning on our own iOS devices showed both audio and video calls, denoted by "Duo Audio" or "Duo Video," in the regular iOS phone app on the "Recents" tab. Interestingly, while audio calls showed up on both sender and receiver iPhones, only sender video calls did.