Use of "to" as a preposition

[Amner]

I have many doubts about this cases (all of them were extracted from books, web pages and so on):

1. 3 steps to losing weight
2. The key to understanding Islam
3. The secret to making money online
4. The Secret to Success (why there's no "ing" like in the previous sentence)

5. 3 steps to lose weight
The requirements to gain entrance . . .


I do not know why it is followed the termination "ing" after "to" when it is used the article "the" at the beggining, but in other cases not even with the same noun after "the" such as "secret" (cases 3 and 4)

Help me out with this too please!
6. We're going to look at a number of approaches to describing the diferrences . . .
7. From playing with sibblings to negotiating allowances

Thanks in advance!

[Cerberus™]

This is a complicated subject. All the -ing forms in your examples are gerunds. There are three general observations I can make.

- A gerund is a sort of noun; a verb that can have a direct object or the object of a preposition can usually also have a gerund instead of a noun.
- Some verbs regularly go with gerunds, others with infinitives (and some can go with either): there is no real rule for this; you'll have to learn which goes with what by practice, or from a dictionary.
- The mistake of using the infinitive where only the gerund is right is often made, but the reverse mistake is not so frequent; when in doubt, I'd go for a gerund.

The sentences you gave are special, since their verbs all have to + gerund. This "to" is here not the infinitive particle, but a regular preposition. In other words: the gerund could in each case be replaced with a noun, as in "three steps to weight loss" or "the key to knowledge of Islam"; the gerund just behaves like any other noun.