A man visiting Al Mamoon is rather flustered after misplacing his diary again.
This errand is available to complete after completing Errand 079.
Speak with the rich man located to your immediate left as you enter the town square of Al Mamoon, and you should recognize him as the traveler from Ding Dong Dell. Looks like he lost his diary again, so off you go.
This time the diary can be found on the trail leading to Old Smoky. It's not too far away actually. Once you pass the first large crack in the ground (on the left side), inspect the big tree just behind it to find the traveler's diary; this is slightly southeast of the first lava stream. Return to the man to complete the errand.