How do I get to the recipe log?

There is no "Recipe Log" in Breath of the Wild.

The closest you can get is by looking at a meal / elixir in your inventory, selecting it, and then selecting "Recipe" (instead of Eat / Drink, if you were consuming it).

This will tell you what ingredients were involved in its creation, even if you originally received the item as a reward (or were otherwise uninvolved in its manufacture).

So the "Recipe Log", as it were, is only as large as the number of cooked items you currently have on you.

The other way to create a "Recipe Log". Is after you create something that you want to remember follow Sable Dreamer Answer's instructions

The closest you can get is by looking at a meal / elixir in your inventory, selecting it, and then selecting "Recipe" (instead of Eat / Drink, if you were consuming it).

Then take a screenshot on the Wii U or the Switch. This allows me to to quickly cycle thru my screenshots to find a Recipe for something I made in the past and is sorta a improvised "Recipe Log".