Logarithms in Basic Math

Introduction

• Logarithms are essential for solving many calculus problems.
• Two critical chapters: Logarithms and Trigonometry.
• Focus on understanding properties and solving problems efficiently.

John Napier

• Inventor of the logarithm concept.
• Prominent quote: Every positive real number (N) can be expressed in exponential form: N = a^L.
• Only positive real numbers are considered for logarithms, not complex numbers.
• Base (a) must be positive and not equal to 1.

Exponential and Logarithmic Forms

• Exponential Form: N = a^L
• Logarithmic Form: L = log_a(N)
• Both forms are equivalent: N = a^L implies L = log_a(N).

Properties of Logarithms

1. Base Change Theorem: Logarithmic change of base

   • log_a(N) = log_b(N) / log_b(a) for any base b > 0 and b ≠ 1.

2. Negative Base Property

   • log_a(1/N) = -log_a(N)
   • Example: log_3(1/9) = -2

3. Sum Property

   • log_a(N1 * N2) = log_a(N1) + log_a(N2)

4. Difference Property

   • log_a(N1 / N2) = log_a(N1) - log_a(N2)

5. Power Property

   • log_a(N^k) = k * log_a(N)
   • Special cases:
     • If k is odd: No absolute value needed
     • If k is even: Absolute value needed

6. Equality and Uniqueness of Logs

   • log_a(a) = 1 and log_a(1) = 0
   • Applying logs: If a^L = N then L = log_a(N).

7. Interchanging Bases and Exponents

   • A^(log_A(N)) = N
   • A^(log_B(N)) = N^(log_B(A))

Simplifying Logarithmic Expressions

• Common Logs (Base 10)

  • Simplify expressions using log(x) for base 10 logs.

• Natural Logs (Base e)

  • Simplify expressions using ln(x) for natural logs with base e.
  • ln(x) = log_e(x).

Solving Logarithmic Equations

1. Prime Factorization Method

   • Break numbers into their prime factors to simplify log expressions.
   • Example: log₂(180), break 180 into prime factors: 180 = 2² * 3² * 5

2. Equating Bases and Exponents

   • Solve equations by changing forms and setting exponents equal.

Common Mistakes

• Confusion in power properties when it's the whole log versus just the argument.
• Correct: log_a(N^k) = k * log_a(N) only if N is to the power k.
• Incorrect: (log_a(N))^k is not the same as the above.*

Example Problems

1. Basic Applications

   • log_2(16) = 4

2. Complex Mixed Problems

   • Including multiple bases and changing logarithmic forms.

Conclusion

• Mastery of logarithms is critical for higher math, including calculus and trigonometry.
• Ensure a strong grasp of properties and practice solving diverse problems.

Homework Problems

• Extensive practice is advised with given homework problems to consolidate learning.