Lecture Notes on Logarithms

Introduction

• Arvind Kalia conducting the session.
• Importance of logarithms for JEE preparation.
• Logarithms are not a formal chapter but a topic under basic mathematics.
• Few properties of logarithms simplify expressions in JEE Main and Advanced.
• Direct questions on logarithms are rare but their properties are used frequently.

Basics of Logarithms

• Definition: log_b(a) is the power c such that b^c = a.
• Example: log_2 8 = 3 because 2^3 = 8.
• Logarithms are defined for positive numbers only.
• log_b(a) for a > 0 and b > 0, b ≠ 1.
• Identity: log_b(b) = 1, log_b(1) = 0.

Important Properties of Logarithms

Basic Properties

1. log_b(MN) = log_b(M) + log_b(N) (Product to Sum)
2. log_b(M/N) = log_b(M) - log_b(N) (Quotient to Difference)
3. log_b(M^k) = k * log_b(M) (Power Rule)

Advanced Properties

1. Base Changing Property: log_b(M) = log_k(M) / log_k(b).
   • Special cases: log_b(M) = 1 / log_M(b).
   • Cross Multiplication Property: log_a(b) * log_c(a) = log_c(b).
2. Power Property in Logarithm: a^(log_b(c)) = c^(log_b(a)).
3. Logarithm of Reciprocal of Base: log_a(b^(n)) = (1/n) * log_a(b).

Solving Logarithmic Equations

• To solve log_b(f(x)) = log_b(g(x)) resulting in f(x) = g(x), ensure the domain is positive.
• For inequations, consider if the base is greater than or less than 1 to determine if the inequality changes direction.
• Example: log_2(x) > log_2(3) implies x > 3 because log base 2 is increasing.
• Special approach: Directly simplify logarithmic terms by keeping the properties in mind (cross multiplication method often helps).

Graphs of Logarithmic Functions

• log_a(x) where a > 1 is an increasing function passing through (1, 0).
• log_a(x) where 0 < a < 1 is a decreasing function passing through (1, 0).

Applications in JEE-type Problems

• Example problem types including finding digits: Number of digits in 2^100 using log`.
• Using properties to simplify and solve logarithmic equations.

Tips and Tricks

1. Always remember the conditions for the argument of a logarithm and log not being defined for non-positives.
2. Use log properties extensively to simplify and reduce complex expressions.
3. For inequations, always consider the behavior of the logarithmic function which depends on whether the base is greater than or less than 1.
4. Practice problems are essential to understand the application of these properties.

Additional Topics Covered

• Graphs of log functions for different bases.
• Inequalities involving logarithms.
• Tricks to solve logarithmic inequalities and understand their domain.

Final Advice

• Practice sheets and specific problem-solving is crucial; theory alone won't suffice.
• Ensure to practice a variety of problems to master the topic.

Summary

• Logarithms are crucial for simplifying many expressions in mathematics, especially for competitive exams like JEE.
• Remember the key properties and practice their applications.
• Use domains and base considerations carefully while solving equations and inequalities.

Resources for Further Practice

• Practice problems and solutions to be referred from the provided sheet.
• Video solutions available for complex problems.