Jeg lægger den anden bog på bordet.

Because Danish distinguishes between:

• at lægge = to lay / put / place something
• at ligge = to lie / be lying

In this sentence, the subject jeg is actively placing the book somewhere, so Danish uses lægger.

Compare:

• Jeg lægger bogen på bordet. = I put the book on the table.
• Bogen ligger på bordet. = The book is lying on the table.

So lægger is the correct present-tense form of at lægge here.

In Danish, when a noun is definite and has an adjective before it, you usually use:

• den/det/de + adjective + noun

So:

• den anden bog = the other book / the second book

If there were no adjective, you would normally use the definite ending directly on the noun:

• bogen = the book

So:

• bogen = the book
• den anden bog = the other/second book

You do not normally say anden bogen in standard Danish.

It can mean either, depending on context.

• second: if you are counting books in order
• other: if you are contrasting one book with another

So den anden bog can mean:

• the second book
• the other book

The surrounding context tells you which meaning is intended.

Because bog is a common-gender noun in Danish:

• en bog = a book

Common-gender nouns use den in this kind of structure:

• den anden bog

If the noun were a neuter noun, you would use det instead:

• et bord = a table
• det store bord = the big table

So den is used because bog is an en-word.

Because Danish usually makes nouns definite by adding an ending to the noun itself.

• et bord = a table
• bordet = the table

So på bordet means on the table.

This is different from den anden bog, where an adjective comes before the noun. In that pattern, Danish uses a separate word (den) before the adjective instead of only adding a definite ending to the noun.

På means on when something is placed on a surface.

Since a book is being put onto the surface of the table, på bordet is the natural choice:

• på bordet = on the table

Compare:

• i bordet = in the table
• under bordet = under the table

So på is used because the book ends up resting on top of the table.

The neutral word order here is:

• Jeg = subject
• lægger = verb
• den anden bog = direct object
• på bordet = place expression

So the sentence pattern is:

Subject + verb + object + place

Also, Danish main clauses follow the V2 rule, which means the finite verb is in the second position in the clause. In this sentence, that happens naturally:

• Jeg lægger den anden bog på bordet.

You can move another element to the front for emphasis, but then the verb still stays second:

• På bordet lægger jeg den anden bog.

That version is more marked or emphatic.

It is the present tense, but Danish present tense is often broader than one single English form.

Depending on context, jeg lægger can correspond to:

• I put
• I am putting
• I place

Danish does not always make the same present/simple-progressive distinction that English does. So the exact English translation depends on the situation.

A rough pronunciation is LEH-ger, but that is only approximate.

A few useful points:

• æ is a front vowel that many English speakers find unfamiliar.
• The g is not pronounced like two separate hard English g sounds.
• The ending -er is weak and unstressed.

In standard Danish, lægger is approximately [ˈlɛɡɐ].

The exact sound is best learned by listening and repeating, because Danish pronunciation is often less phonetic than English speakers expect.

Here bog is singular and in its basic noun form.

Its main forms are:

• en bog = a book
• bogen = the book
• bøger = books
• bøgerne = the books

In den anden bog, the noun stays in that basic singular form bog because the definiteness is shown by den plus the adjective pattern.