Logarithms: In Details

Introduction

• Discussion on the importance of logarithms
• Covering both complex and simple examples
• Agenda: Basics, properties, extra topics (graphs and inequalities)
• Main objective: Preparation for JEE

Basic Definitions

• Log of b to the base a = c
  • If a^c = b
  • Example: Log of 8 base 2 = 3
• Special features:
  • Log of a base a = 1
  • Log of 1 base a = 0

Properties of Logarithms

• Main property:
  • Addition of log: Log m + Log n = Log (m * n)
  • Subtraction of log: Log m - Log n = Log (m / n)
  • Power property: n * Log m = Log (m^n)

Solved Examples

1. Basic log calculation

• Log 100 base 10 + Log 1000 base 10 = Log 100000 base 10 = 5

2. Addition and subtraction

• Example: Log of 15 - Log of 5 = Log (15/5) = 1

3. Power property

• Example: 2 * Log x base 10 = Log (x^2) base 10*

Solving Log Variables

• Logs are like boys and girls (one-to-one base comparison)
• If log base is greater than 1, the inequality value does not change
• If log base is less than 1, the inequality value changes

Graphical Interpretation

• Log function graph
  • When base is greater than 1, the graph increases
  • When base is less than 1, the graph decreases
• Important points: Basic properties of log and their graphs

Logarithm Inequalities

• If base is greater than 1, the log inequality does not change
• If base is less than 1, the log inequality changes
• Solving:
  • Taking log on both sides or equating
  • Finding intersection of the common part

Base Change Property

• How to convert log to a different base
• Properties:
  • If log b base a = c, then log b/log a = c
• Usage: Simplifying sums and questions

Special Cases

• Log b / Log a = Log b to the base a
• Formula: Log a base a = 1

Practice Problems

• New problems to help understand properties
• Solve all types of problems

Pro Tips

• Read questions carefully
• Solve according to their properties
• Keep a good grasp on basic math
• Focus on graphs and inequalities

Subscription Information

• On-demand courses: Batch of teachers A, B, C, D
• One month subscription: Available at $9499
• An important option to prepare yourself
• Once subscribed, practice becomes very easy
• Different batches available