Við athugum persónuafsláttinn í ljósi þess að vaxtagjöldin eru forskráð.

Because athugum is the 1st person plural present tense form of the verb athuga. The subject við means we, and the verb ending -um matches that subject.

So:

• að athuga = to examine / check / consider
• við athugum = we examine / we check

Icelandic verbs usually show person and number more clearly than English verbs do.

It is the direct object of athugum, so it appears in the accusative case.

The base noun is:

• persónuafsláttur = personal tax credit / personal allowance

This noun is masculine singular. In this sentence it is also definite: the personal tax credit. That gives the form:

• nominative singular indefinite: persónuafsláttur
• accusative singular indefinite: persónuafslátt
• accusative singular definite: persónuafsláttinn

So the ending -inn here is the suffixed definite article, and the whole form reflects accusative singular definite masculine.

Because Icelandic very often puts the definite article onto the end of the noun instead of using a separate word like English the.

So in this sentence:

• persónuafsláttinn = the personal tax credit
• vaxtagjöldin = the interest expenses

The exact form of that ending changes depending on gender, number, and case. That is why you see -inn in one noun and -in in the other.

It is a fixed expression meaning something like:

• in light of the fact that
• given that
• since
• considering that

It is a fairly formal or administrative way to introduce a reason or relevant circumstance.

So:

• í ljósi þess að vaxtagjöldin eru forskráð
  means roughly
• in light of the fact that the interest expenses are pre-registered

Because þess is the genitive form of það, and this phrase requires that form.

The expression is built like this:

• í ljósi e-s = in light of something

Here e-s means the following word is in the genitive. So:

• í ljósi þess = in light of that

Then að introduces the clause:

• í ljósi þess að... = in light of the fact that...

So þess is there because of the grammar of the expression, not because it means something different from það in a basic vocabulary sense.

Vaxtagjöldin is a compound noun meaning the interest expenses / the interest payments.

You can break it down like this:

• vaxta- = interest-
• gjöld = fees, charges, expenses

The ending -in is the suffixed definite article here. This noun is neuter plural, and in this sentence it is the subject of the clause, so it is in the nominative plural definite form:

• vaxtagjöld = interest expenses
• vaxtagjöldin = the interest expenses

Because vaxtagjöldin is plural, and eru is the plural present tense form of vera (to be).

So:

• er = is for singular
• eru = are for plural

Since vaxtagjöldin means the interest expenses, the sentence needs eru:

• vaxtagjöldin eru... = the interest expenses are...

Forskráð is the past participle / adjective from forskrá meaning to pre-register. In the phrase eru forskráð, it means are pre-registered.

It does agree with the subject, but the neuter plural form happens to look the same as the neuter singular form:

• masculine plural: forskráðir
• feminine plural: forskráðar
• neuter singular: forskráð
• neuter plural: forskráð

Since vaxtagjöldin is neuter plural, forskráð is exactly the correct agreeing form.

Yes, but this sentence is fairly straightforward.

The main clause is:

• Við athugum persónuafsláttinn...

That is a normal Icelandic main-clause order: subject + finite verb + object.

Then the sentence continues with a subordinate clause introduced by að:

• ...að vaxtagjöldin eru forskráð

Inside that subordinate clause, the order is also very normal:

• subject + verb + complement

So this sentence is useful because it shows standard Icelandic word order without anything especially unusual.

A very helpful strategy is to break them into parts. Icelandic uses compounds constantly.

For example:

• persónuafsláttur
  = persónu-

  • afsláttur
    = literally something like person discount/allowance, here personal tax credit

• vaxtagjöld
  = vaxta-

  • gjöld
    = interest expenses

• forskráð
  = for-

  • skráð
    = pre-registered

A useful rule is that the last part of a compound usually determines the basic grammar of the word, such as its gender and inflection pattern. That makes long words much less intimidating once you start recognizing the pieces.