Advanced Logarithmic Problem Solving

This page presents a challenging logarithmic problem and its solution, showcasing advanced techniques in logarytmy - zadania z rozwiązaniami. The problem involves calculating a complex logarithmic expression with base 3.

The main equation to be solved is:

log₃(30 · 3³) - log₃(√(1093 · 243)) + log₃(3⁻⁶⁰⁹³)

The solution process demonstrates several key concepts in logarithmic manipulation:

Highlight: The problem combines logarithms with the same base (3) but different arguments, requiring the application of logarithm properties for simplification.

Example: log₃(30 · 3³) is simplified by using the product rule of logarithms and the power property.

Vocabulary: Base of a logarithm - In this case, 3 is the base for all logarithms in the problem.

The solution involves the following steps:

1. Simplifying log₃(30 · 3³) using logarithm properties
2. Calculating log₃(√(1093 · 243)) by applying the square root property of logarithms
3. Simplifying log₃(3⁻⁶⁰⁹³) using the power property of logarithms
4. Combining the simplified terms

Definition: Product rule of logarithms: log₃(a · b) = log₃(a) + log₃(b)

The final answer is presented as a fraction: 27/12. This result demonstrates the importance of precise calculations and the application of logarytmy wzory in solving complex logarithmic problems.

Quote: "Odpowiedź: Wynik to 27/12" (Answer: The result is 27/12)

This problem serves as an excellent example for students preparing for advanced mathematics exams, particularly those focusing on logarytmy - zadania maturalne rozszerzenie. It showcases the type of complex reasoning and multi-step problem-solving required in higher-level mathematics courses.