Let \(m\) be the minimum possible value of \(\log _{3}\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right)\), where \(y_{1}, y_{2}, y_{3}\) are real numbers for which \(y_{1}+y_{2}+y_{3}=9\) .Let \(M\) be the maximum possible value of \(\left(\log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}\right)\), where \(x_{1}, x_{2}, x_{3}\) are positive real numbers for which \(x_{1}+x_{2}+x_{3}=9\) .Then the value of \(\log _{2}\left(m^{3}\right)+\log _{3}\left(M^{2}\right)\) is_______
Correct Answer: 8
Solution and Explanation
\(\text{The Value of}\) \(\log_2(m^3)+\log_3(M^2)\) \(is\;\underline{8}.\)
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Concepts Used:
Determiners
Determiners are words that are used to introduce and specify nouns in a sentence. They provide information about the number, ownership, quantity, and definiteness of the noun. In just 200 words, let's explore the key aspects and usage of determiners.
Determiners include articles (a, an, the), demonstratives (this, that, these, those), possessives (my, your, his, her, its, our, their), quantifiers (some, any, few, several, many, much), numbers (one, two, three), interrogatives (which, what), and distributives (each, every, either, neither).
Articles:
- Definite Article: "The" is used to refer to a specific noun, such as "the book" or "the car."
- Indefinite Articles: "A" and "an" are used to refer to a non-specific noun, such as "a book" or "an apple."
Demonstratives:
- These determiners point to specific objects or people. For example, "this car" or "those books."
Possessives:
- These determiners indicate ownership or possession. For instance, "my house" or "their dog."
Quantifiers:
- Quantifiers provide information about the quantity or amount of the noun. For example, "some water" or "many books."
Numbers:
- Numbers are used to indicate the exact quantity of a noun. For instance, "three cats" or "ten students."
Interrogatives:
- These determiners introduce questions and seek information. For example, "which color" or "what time."
Distributives:
- Distributives refer to individual members of a group. For instance, "each student" or "every child."
Determiners play a crucial role in specifying and clarifying nouns in a sentence. They help convey information about the noun's definiteness, ownership, quantity, or question being asked. Correct usage of determiners is essential for clear and precise communication.
It is important to note that determiners are used at the beginning of noun phrases and are followed by the noun they modify. They agree with the noun in terms of number and gender.
Overall, determiners provide important contextual information about nouns and contribute to the overall meaning and structure of a sentence. Their accurate use enhances clarity and specificity in communication.