Basic Mathematics Lecture on Logarithms

Importance and Application

• Logarithms are crucial in calculus for solving mixed problems involving logs.
• Important chapters: Logarithm and Trigonometry.
• Mastery of these chapters helps in solving function problems in calculus.

Introduction to Logarithms

• John Napier: Introduced the concept of logarithms.

• Key Idea: Every positive real number (N) can be expressed in exponential form as:

  $$N = a^L$$

  where:

  • N: Positive real number
  • a: Base (a > 0 and a ≠ 1)
  • L: Exponent or logarithm

Properties of Logarithms

1. Logarithm as Power Representation: $$L = log_a(N)$$ $$N = a^L$$

2. Change of Base Formula:

   $$log_a(N) = \frac{log_{\alpha}(N)}{log_{\alpha}(a)}$$

3. Special Base Properties:

   • Common log: $$log_{10}(x)$$
   • Natural log: $$log_e(x)$$ or $$LN(x)$$, where 'e' is Napier's constant (~2.718).

4. Logarithmic Identities:

   • Product rule:

     $$log_a(N_1 imes N_2) = log_a(N_1) + log_a(N_2)$$

   • Quotient rule:

     $$log_a\left(\frac{N_1}{N_2}\right) = log_a(N_1) - log_a(N_2)$$

   • Power rule (when n is odd or even):

     $$log_a(N^n) = n \times log_a(|N|)$$ (Use modulus |N| when n is even)

5. Logarithmic and Exponential Relationship:

   $$a^{log_a(N)} = N$$

6. Reciprocal Property:

   $$log_a\left(\frac{1}{N}\right) = -log_a(N)$$

7. Logarithm of the Base Itself:

   $$log_a(a) = 1$$

8. Logarithm of One:

   $$log_a(1) = 0$$

9. Symmetry Property:

   $$log_a\left(N^B\right) = B \times log_a(N)$$

   • For example,

     $$log_a(N^2) = 2 \times log_a(N)$$

   • This property avoids common mistakes like assuming the entire logarithm to a power.

Solving Logarithmic Equations

1. Given:

   $$log_2(x) = 5$$

   • Rewrite in exponential form: $$2^5 = x$$
   • Solution: $$x = 32$$

2. For more complex expressions with mixed logarithmic forms, use appropriate properties to simplify.

Practice and Homework

1. Practice Problems:
   • Simplify expressions like $$log_{10}(100)$$, $$log_3(27)$$, $$log_5(25) - log_5(5)$$.
2. Advanced Problems:
   • Solve logarithmic equations and validate the properties.

Conclusion and Remarks

• Revise properties and ensure understanding to tackle calculus problems effectively.
• Next topic: Set Theory and discussion on previous homework questions.