Motivation
I debated this question recently with my girlfriend and my flatmate. As a mathematics student, who looked into a bunch of languages, I wanted to share my thoughts.
Disclaimer
Honestly, I am not entirely sure the question is understood the same way in English as the original German question: „Sind Sprachen logisch?“. Depending on your understanding, the word “logical” yields different associations. I assume this is the best possible translation for now.
Preliminaries
Language and structures
Languages usually consist of a vocabulary and grammar in its simplest form. Natural languages, in particular, have grown over years and many people with different mindsets contributed to the language. Three examples:

German (like many languages) is more and more influenced by English terminology which follows a different mindset (e.g. should jeans be a plural or singular as anglizism?).

In Hebrew, vowels are skipped traditionally. However in Modern Hebrew, כתיב חסר ניקוד (engl. full spelling) is becoming more popular and in the case of Mater lectionis spelling, consonants are used to indicate a vowel. This introduces a different mindset to resolve ambiguities.

In 633–654 CE, the Islamization of Iran took place and as a result, arabization of the Persian language took place. This results in 24% of the vocabulary of an everyday vocabulary of 20,000 words in current Persian being of Arabic origin (according to John R. Perry on Wikipedia).
As a result, natural language often incorporates various structures the native speaker might not even be familiar with.
Does a language without structure exist?
 No.

This would mean that any message, you want to convey, gets its own symbol without association to any other symbol. Think of it as living for one day and enumerate every thought you have. If you want to convey the same message again, you need to recall the number you used for this thought. I am not familiar with any person with such memory capacity. It is not practical in any way.
 It depends.

You might want to claim that “structure” is a manmade concept. As such it depends on your definition. Maybe in the previous example, you might recognize (contrived example following) that only messages with prime numbers refer to yourself. It this structure?
What is logic?
Logic is the systematic study of valid rules of inference, i.e. the relations that lead to the acceptance of one proposition (the conclusion) on the basis of a set of other propositions (premises). More broadly, logic is the analysis and appraisal of arguments. —Wikipedia: Logic
Mathematics and computer science define various kinds of logic. Fuzzy logic, temporal logic and firstorder logic shall be given for reference. They do not necessarily have anything in common. The foundation (for most of mathematics) is the axiom set by ZermeloFraenkel in combination with the the controversial Axiom of Choice (⇒ ZFC). Because formally ZFC is a onesorted theory in firstorder logic, most mathematical theorems rely upon firstorder logic.
However, in computer science, you will often find that propositional logic is used as basis (due to computability issues). This subset of firstorder logic is also, what people are more familiar with.
What characterizes logic? Logic is defined by a set of axioms that allow to infer new knowledge. In propositional logic, every statement is true or false. If we negate “A and B”, then we get “(not A) or (not B)”. This is stated by De Morgan’s laws. Propositional logic defines the axioms (true/false, and/or/not operators) and the inference rules are to be discovered. In the end, which axioms you define, is up to you. I have not yet discovered a math paper which defines something “for fun”, but because it is “useful” to discover new structures and relations.
Examples
A simple example for structure
I cannot go to school.
I can’t go to school.
We can derive a simple rule:
If “not” occurs in a compound word, “not” can be replaced by “n’t”.
Dealing with ambiguity – a counterexample of structure?
Do you want coffee or tea?
Do you want coffee and tea?
From a logicians’s perspective, “want coffee or tea” is true if and only if “want coffee” or “want tea” (any of the two or both) is true. This is different from “want coffee and tea” being true if and only if “want coffee” and “want tea” is true (both of the two). If my friend asks me the question upon visit, I will understand the question the same way. I will chose tea, coffee, or in an unusual case both. So from a logical point of view (POV), there is a difference but from a linguistic POW, there is none.
This is ambiguity. This is nonconsistency. But is this a counterexample for structure? I think so, since the choice of word is significant in other context:
I will watch a video or read a book.
In this case, the speaker certainly wants to convey that he/she is going to pick one of them, but not both. Exchanging “or” with “and” would specify a sequence of actions.
Exceptions violate logic?
There are exceptions in every natural language. You might have a harder time finding one in the vocabulary, because few people are into etymology, but everyone knows exceptions in pronunciation of words (think of loanwords, for example).
Let’s have a look at our previous rule “If ‘not’ occurs in a compound word, ‘not’ can be replaced by ‘n’t’”. This will be a contrived example, but it establishes an exception. The word notable could be considered as compound word (“not” and “able”). However, if we apply the rule, it yields “n’table” and “notable” as admissible words. But the former is just wrong. We could rephrase our rule (“If ‘not’ occurs in a compound word and refers to semantic negation, ‘not’ can be replaced by ‘n’t’”) or we could add an exception:
If “not” occurs in a compound word, “not” can be replaced by “n’t”; except for “notable”.
What might surprise people is that this is valid, not only from a linguistic POV, but also from a logical POV. Exceptions do not violate logic. In many mathematical theorem, you will find statements like “any, but not the same” and “greater but not equal” which model exceptions.
Summary

Any natural language has structure.

Mathematics allows you to define any axioms establishing your own kind of logic.

Ambiguities and contextsensitivity violate structure.

Exceptions do not violate structure.
So, where to go with this? Can we just collapse the words “structure” and “logic”? Some thoughts:

Any natural language is structured.

You can always define your own axioms. You can always define your own inference rules. You can always claim X is logic.

Ambiguities and contextsensitivity must be eliminated by assigning a specific meaning to every structure (ability to do so might be a research question on its own)

All exceptions must be included in the axioms of the logic.
So we could claim the structure of a natural language is logic. But consider “structure” as counterterm to “logic”. Because what happens if we collapse these words?

Mathematicians will ignore it (diplomatic approach) or not accept it (pragmatic approach).

Unless you reach consistency or completeness, your logic is not valuable for mathematics.

Writing down all axioms of a language, is an intriguing task for computational linguistic, but I cannot see any results in the near future.

I expect the axioms of a natural language to be a myriad of exceptions and not helpful to (e.g.) study a new language.
As a result, it is helpful to distinguish logic (axioms, inference rules, complete or consistent) and structure (rules of thumb which hold  at least mostly  true).
Conclusion
Natural languages are not logical in the mathematical sense, because claimed axioms do not reach consistency or completeness at all. Current state of the art is that the axioms of any natural language cannot be expressed explicitly, rendering the language as logic as meaningless to mathematics. However,

“logical” is a colloquial expression for “has some inherent structure”. In this context, “natural language are logical” indeed.

Language changes over time. If you actually find the set of axioms of any language, please tell me and don’t forget to update it at least yearly.