Chapter 2: 9th Class Math Notes

Chapter 2: 9th Class Math Notes are available for students with step-by-step instructions. This chapter is about logarithms.

This chapter requires knowledge of real numbers, including logarithms, scientific notation, number representation, and some techniques to solve the variables.

Chapter 2: 9th Class Math Notes

Class 9 Math Notes – Chapter 2 Summary

This summary covers key concepts and solution methods from Chapter 2 of your 9th class mathematics notes:

Scientific Notation

• Scientific Notation: Expressing numbers in the form a×10ⁿ, where 1≤a<10 and n is an integer.
  • Examples:
    • 2000000=2×10⁶
    • 0.0000009=9×10⁻⁷
  • Ordinary Notation: Converting back from scientific notation.
    • 8.04×10²=804

2. Logarithmic and Exponential Forms

• Logarithmic Form: log⁡_b(x)=ylog_b​(x)=y means b^y=x.
  • Examples:
    • 10³=1000 → log⁡₁₀1000=3
    • 2⁸=256→ log⁡₂256=8
• Exponential Form: Converting logarithmic equations back to exponential form.
  • Examples:
    • log⁡₅125=3 → 5³=125
    • log⁡216=4log2​16=4 → 24=1624=16

Solving Logarithmic Equations

• Finding x:
  • log_⁡x64=3 → x³=64→ x=4
  • log⁡₁₀x=−3 → x=10⁻³=0.001

Characteristics and Mantissa

• Characteristic: The integer part of a logarithm.
  • Examples:
    • log⁡43=1.6335 → Characteristic = 1, Mantissa = 0.6335
    • log⁡0.0876=−2+0.9425 → Characteristic = -2, Mantissa = 0.9425

Laws of Logarithms

• Product Rule: log⁡b(xy)=log⁡_b(x)+log_⁡b(y)
• Quotient Rule: log⁡_b(xy)=log⁡_b(x)−log_⁡b(y)
• Power Rule: log⁡b(xⁿ)=nlog⁡b(x)
• Examples:
  • log⁡₂18−log⁡₂9=log⁡₂(18/9)=log⁡₂2=1
  • 2log⁡2+log⁡25=log⁡2²+log⁡25=log⁡(4×25)=log⁡100=2

Applications of Logarithms

• Solving Exponential Equations:
  • (81)^x=(243)^x+2 → 3^4x=3^5x+10 → x=−10
• Population Growth:
  • P(t)=22(1.025)^t → Solve for t when P(t)=35 →t≈19 years (2035)

Logarithm Tables

• Using Logarithm Tables:
  • log⁡(3.68×4.21/5.234)=log⁡3.68+log⁡4.21−log⁡5.234=0.4713 → Antilog = 2.960

Review Exercise

• Multiple Choice Questions:
  • log⁡₂2³=3
  • log⁡100=2
  • log⁡(0) is undefined
• Expressing in Logarithmic/Exponential Forms:
  • 3⁷=2187 → log⁡₃2187=7
  • log⁡₄8=x → 4^x=8

9th Class Math Notes Full Book