Koine Greek is a highly inflected language.1 With respect to substantives,2 this means that the ending of a substantive will change in spelling in relation to its case. This is called declension.3 Bullions defines case as “the state or condition of a noun with respect to the other words in a sentence.”4 Koine Greek is understood to have five cases: “the Nominative, Genitive, Dative, Accusative, and Vocative.”5
In a lexicon, words are generally arranged alphabetically according to lemma.6 The lemma of a substantive is its form as declined in the nominative case and singular number. Hence, one would find the word λόγος as a lemma in a lexicon.
A substantive declined in the nominative case is generally used as the subject of a sentence. It could also, among other things, be a predicate nominative, functioning as a vocative, etc.7
The genitive case has a variety of functions, by far more than the nominative. In 1 Peter 1:23, the word λόγου is declined in the genitive case because λόγου (the object of the preposition διὰ) is being governed by the preposition διὰ (διά).8 The preposition διά followed by a substantive in the genitive case can have several meanings, one of which is to indicate the means by which something is done.
On the preposition διά, Thayer notes,9
Hence, διὰ λόγου can be understood as “by the word” or “by means of the word.” The preposition διά also governs objects of the preposition declined in the accusative, but that is not the case here, as we would instead read λόγον (accusative) instead of λόγου (genitive).10
In 1 John 1:1, the word λόγου occurs again, but this time, it is the object of the preposition περὶ (περί).
On the preposition περί, Thayer notes,11
Hence, περὶ τοῦ λόγου can be understood as “about the word.” That is, “we have heard...about the word.”
What’s the difference between λόγος and λόγου?
Semantically, there is no difference in meaning; they are simply different cases of the same lexeme. However, syntactically, they differ as the subject of a sentence and the object of a preposition, respectively.
Long story short, introduce yourself to Koine Greek by investing in a biblical Greek grammar.12 You can return and re-read my answer. The cloud will have removed itself by then, and things will be clearer.
Footnotes
1 Black, Ch. 2, §12, p. 10
2 Aaron & Fowler, p. 902: “a word or word group used as a noun.” In Koine Greek, adjectives and participles may also function substantively (i.e., as nouns).
3 When a substantive inflects, it is said to decline. When a verb inflects, it is said to be conjugated. According to Wikipedia on “inflection”:
The inflection of verbs is called conjugation, and one can refer to the inflection of nouns, adjectives, adverbs, pronouns, determiners, participles, prepositions and postpositions, numerals, articles etc., as declension.
4 Bullions, §11.3, p. 15
5 id., p. 16
6 Oxford English Dictionary, “lemma,” 2. c.,: A lexical item as it is presented, usually in a standardized form, in a dictionary entry; a definiendum.
7 Bullions, §137, p. 201
8 Oxford English Dictionary, “govern,” 13.: transitive. Grammar. Of a word, esp. a verb or a preposition: to have (a word or a case) depending on it; to require (a certain case or mood) in a dependent word.
9 Thayer, p. 132–133
10 id., p. 501
11 cf. Matt. 13:21: διὰ τὸν λόγον
12 William D. Mounce’s Basics of Biblical Greek is a decent introductory grammar of biblical/Koine Greek.
References
Aaron, Jane E.; Fowler, H. Ramsey. The Little Brown Handbook. 11th ed. London: Pearson, 2010.
Black, David Alan. Learn to Read New Testament Greek. 3rd ed. Nashville: B&H Publishing Group, 2009.
Bullions, Peter. The Principles of Greek Grammar: Comprising the Substance of the Most Approved Greek Grammars Extant, for the Use of Schools and Colleges. 21st ed. New York: Pratt, 1851.
OED Online. Oxford University Press.
Wilke, Christian Gottlob. A Greek-English Lexicon of the New Testament: Being Grimm Wilke’s Clavis Novi Testamenti. Trans. Thayer, Joseph Henry. Ed. Grimm, Carl Ludwig Wilibald. Rev. ed. New York: American Book, 1889.