Learn Hardcore Japanese: syukudai wo wasurenai you ni, nikki ni kakimasu. - I write it in my diary so that I don't forget my homework.

If you used 忘れるように, it would mean “so that I forget,” which is the opposite meaning.
So we must use the negative form 忘れない before ように here.

What exactly does ように mean in this sentence?

ように after a verb often means:

“so that / in such a way that / in order that”

In this sentence:

忘れないように = “so that I don’t forget”
宿題を忘れないように、日記に書きます。
= “(I) write it in my diary so that I don’t forget my homework.”

Grammatically, it links a purpose/result clause (忘れない) to the main action (書きます).

How is ように different from ために for “in order to”?

Both can translate as “in order to,” but they have different typical uses.

V-る + ために: used mainly with volitional actions (things you actively do on purpose).
- e.g. 日本語を勉強するために、日本へ行きました。
  “I went to Japan in order to study Japanese.”
V-る / V-ない + ように: often used when:
- talking about a result you want (not directly controllable), or
- talking in a softer, less direct way.

In 宿題を忘れないように、日記に書きます。, 忘れない is a state/result (not forgetting), so ように sounds more natural than ために.
You could say 忘れないために, and it’s not wrong, but ように is more common and feels softer.

Why is the verb before ように in plain form (忘れない) and not polite (忘れません)?

Before ように, verbs normally appear in the plain (dictionary) form, even in polite sentences.

Correct: 忘れないように、日記に書きます。
Incorrect: 忘れませんように、日記に書きます。 (sounds wrong in ordinary speech)

So the pattern is:

(plain form) + ように、(polite or plain main clause)

For example:

遅れないように、早く家を出ます。
“I leave home early so that I won’t be late.”

Why is it 日記に書きます and not 日記を書きます?

書く can take different particles depending on what you focus on:

Xを書く – X is the content you write.
- 手紙を書く = “write a letter”
- レポートを書く = “write a report”
Xに書く – X is the place or medium where you write.
- ノートに書く = “write in a notebook”
- 黒板に書く = “write on the blackboard”
- 日記に書く = “write in (a) diary”

Here, the important point is where you write (in the diary), not what the diary is as content. So 日記 is treated as the target/location of writing and takes に.

There is no “I” or “my” in the Japanese sentence. How do we know it means “I write (my homework) in my diary”?

Japanese usually omits the subject and possessives when they are obvious from context.

(私は) 宿題を忘れないように、日記に書きます。
- Subject 私 (“I”) is dropped.
宿題 is naturally understood as “my homework” in a context where you’re talking about yourself.
日記 is similarly understood as “my diary”.

So the full English translation has “I” and “my,” but they’re not stated in Japanese because they’re inferred.

Why is the verb 書きます in present/future tense? How does that work here?

Japanese has basically two tenses:

non-past (書く / 書きます) – covers present and future, and also habitual actions.
past (書いた / 書きました)

Here, 書きます expresses a habitual action / regular behavior:

“I (generally) write it in my diary (as a habit) so that I don’t forget.”

So the non-past polite form naturally means “I do this (regularly)” or “I will do this (as my way of dealing with it).”

Can I reverse the order and say 日記に書きます、宿題を忘れないように。?

You can put the ように-clause after the main clause, and it is grammatically possible:

日記に書きます。宿題を忘れないように。

However:

The most natural and common pattern is:
(reason/purpose clause) + ように、(main action)
→ 宿題を忘れないように、日記に書きます。

If you put it after, it can sound a bit like an afterthought, or more like an explanation added later. For textbook-style, clear Japanese, keep 忘れないように first.

Could I say 宿題を忘れないで、日記に書きます。 instead of 忘れないように?

No, that would change the meaning.

V-ないで most often means “without doing V” or can be used in commands / requests (“Don’t do V and …”).
- 宿題を忘れないで、日記に書きます。
  tends to sound like: “Without forgetting the homework, I write it in my diary,” or like you’re telling someone “Don’t forget your homework, and write it in your diary,” which is not the intended sense.

For expressing purpose (“so that I don’t forget”), the natural form is:

V-ないように、…
→ 宿題を忘れないように、日記に書きます。