Detailed Notes on Logarithms: NCERT References and Key Concepts

1. Definition and Properties of Logarithms

• Definition: Logarithm of a number (x) to the base (b), denoted as (\log_b(x)), is the exponent to which (b) must be raised to obtain (x).

Reference: NCERT Class 12, Chapter 3: Logarithmic Functions

• Properties of Logarithms:

1. Multiplication Rule: (\log_b(xy) = \log_b(x)+\log_b(y))

2. Quotient Rule: (\log_b\left(\frac{x}{y}\right) = \log_b(x)-\log_b(y))

3. Power Rule: (\log_b(x^n) = n \cdot \log_b(x))

4. Logarithm of 1: (\log_b(1) = 0)

5. Logarithm of Base: (\log_b(b) = 1)

6. Change of Base Formula: (\log_b(x) = \frac{\log_a(x)}{\log_a(b)}, a,b>0, a\ne1, b\ne1)

7. Natural Logarithm: (\log_{10}(x)) is the common logarithm, while (ln(x) ) is the natural logarithm with the base (e), where (e) is approximately 2.71828.

Reference: NCERT Class 12, Chapter 3: Logarithmic Functions

2. Applications of Logarithms

• Simplifying and Solving Exponential Expressions: Logarithms are used to simplify and solve exponential equations by converting them into linear equations.

Reference: NCERT Class 12, Chapter 3: Logarithmic Functions

• Calculus: Logarithmic functions and their derivatives and integrals are useful in studying the behavior and properties of various mathematical functions.

Reference: NCERT Class 12, Chapter 9: Differential Equations

• Scientific and Real-World Applications: Logarithmic functions are used in various scientific and real-world applications, including pH calculations, radioactive decay, sound decibel levels, and more.

Reference: NCERT Class 12, Chapter 14: Mathematical Reasoning

3. Inverse Functions

• Logarithmic and exponential functions are inverse functions of each other. If (y = log_b(x)), then (x = b^y).

Reference: NCERT Class 12, Chapter 3: Logarithmic Functions

4. Change of Base Formula

• To convert the logarithm of a number from one base to another, use the following formula:

$$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}, where a,b>0, a\ne1, b\ne1$$

Reference: NCERT Class 11, Chapter 13: Logarithms

5. Logarithmic Equations

• Solving logarithmic equations involves isolating the logarithm term and applying algebraic techniques to simplify and find the value of the variable.

Reference: NCERT Class 12, Chapter 3: Logarithmic Functions

6. Logarithmic Functions: Graphing and Properties

• Logarithmic functions are a family of functions defined by the equation (f(x) = log_b(x)), where (b>0) and (b \ne 1).

Properties of Logarithmic Functions:

1. Domain and Range: Domain is ((0, \infty)) and the range is ((-\infty, \infty)).

2. Increasing and Decreasing: Logarithmic functions are increasing for (b > 1 ) and decreasing for (0 < b < 1 ).

3. Asymptotes: Logarithmic functions have a vertical asymptote at (x=0).

4. Intercepts: ((1,0)) is the only (x)-intercept for all logarithmic functions.

Reference: NCERT Class 12, Chapter 3: Logarithmic Functions

By comprehending these detailed notes and thoroughly practicing problems from various sources, including the NCERT textbooks and past exam papers, you will strengthen your understanding of the concept of logarithms and enhance your preparation for the JEE exam.