Logarithms

A logarithm is a mathematical concept that represents the exponent to which a base must be raised to produce a given number. In simple terms, logarithms are the inverse operations of exponentiation. For example, since 10² = 100, then log₁₀ 100 = 2.

In this article we are going to learn about the history, types, formulas, and logarithm examples, and understand why they are so important in modern mathematics and science.

 

Table of Contents

• History of logarithm
• What are Logarithms?
• Types of Logarithms
• Properties of Logarithms
• Logarithm Formulas
• Logarithm Examples
• Practice Questions
• Real-Life Applications
• Conclusion
• FAQs on Logarithm

 

History of Logarithms

The concept of logarithms was introduced in the 17th century by John Napier. They quickly became an essential tool in science and navigation, helping simplify complex calculations before the invention of calculators.

 

What are Logarithms?

 A logarithm tells us how many times a base is multiplied by itself to reach a given number.
The general form is:
log₍b₎ x = n ⇔ bⁿ = x

Where:

• x is the argument (the number we want to reach)

• b is the base (must be a positive real number ≠ 1)

• n is the exponent or power
  
  

Example:
log₍3₎(27) = 3 → Since 3³ = 27

 

Types of Logarithms

Common Logarithm (Base 10)
Notation: $log₁₀(x) or simply log(x)$
Example: log(1000) = 3 because 10³ = 1000

Natural Logarithm (Base e)
Notation: $ln(x) or logₑ(x)$

e ≈ 2.71828 (Euler’s number)
Example: ln(78) ≈ 4.357

 

Properties of Logarithms

1. Product Rule
   $log₍b₎(mn) = log₍b₎m + log₍b₎n$

2. Quotient Rule
   $log₍b₎(m/n) = log₍b₎m − log₍b₎n$

3. Power Rule
   $log₍b₎(mⁿ) = n × log₍b₎m$

4. Change of Base Rule
   $log₍b₎m = log₍a₎m / log₍a₎b$

5. Base Switch Rule
   $log₍b₎a = 1 / log₍a₎b$

6. Derivative of log
   If  $f(x) = log₍b₎(x)$, then
   f'(x) = 1 / (x × ln(b))

7. Integral of log
   $∫log₍b₎(x) dx = x(log₍b₎x − 1/ln(b)) + C$

8. Other Properties
   $log₍b₎(b) = 1$        $log₍b₎(1) = 0$
   
   $log₍b₎(0) = undefined$

 

Logarithm Formulas

• $log₍b₎(mn) = log₍b₎(m) + log₍b₎(n)$

• $log₍b₎(m/n) = log₍b₎(m) − log₍b₎(n)$

• $log₍b₎(xʸ) = y × log₍b₎(x)$

• $log₍b₎(√n) = (1/2) × log₍b₎(n)$

• $m log₍b₎(x) + n log₍b₎(y) = log₍b₎(xᵐyⁿ)$

 

Logarithm Examples

Example 1:
log₂(64) = ?
→ 2⁶ = 64 → Answer: 6

Example 2:
log₁₀(100) = ?
→ 10² = 100 → Answer: 2

Example 3:
log₃x = log₃4 + log₃7
→ By product rule: log₃x = log₃(28)
→ x = 28

Example 4:
log₂x = 5
→ Convert to exponential: 2⁵ = x → x = 32

Example 5:
log₅(1/25) = log₅1 − log₅25 = 0 − 2 = –2

 

Practice Questions

• Evaluate: log₄(64)

• Solve: log₂x = 3

• Simplify: log₅(25) + log₅(4)

• Find x: $log₃x = log₃9 + log₃3$

• Differentiate:$ f(x) = logₑ(x² + 1) $

Real-Life Applications

• Earthquake Measurement (Richter scale):
  Logarithms help measure the strength of earthquakes on a scale that compares energy released.
• Sound Intensity (Decibels):
  Logarithms are used to calculate how loud a sound is, making it easier to handle very large or small sound levels.
• pH Level in Chemistry (pH = –log[H⁺]):
  Logarithms help measure the acidity or alkalinity of a solution in a simple number.
• Radioactive Decay Calculations:
  Logarithms are used to find how long it takes for radioactive substances to reduce to a certain amount.
• Financial Models and Algorithms:
  Logarithms help model growth, interest, and risk in finance, simplifying complex calculations.
• Data Compression and Algorithm Complexity:
  Logarithms are used in computer science to understand data sizes and how fast algorithms grow or shrink.

 

Conclusion 

Logarithms are foundational tools in mathematics used for simplifying complex calculations. From representing large numbers efficiently to real-world applications in science and engineering, they are essential to understand and master. Start with the basic rules and properties and practice real-world logarithm examples to gain confidence.

 

Frequently Asked Questions on Logarithms

1. What are logarithms?

Ans. Logarithms express the exponent needed for a base to reach a given number. For example, log₁₀(100) = 2.

2. What are the two types of logarithms?

 Ans.Common logarithm (base 10) and natural logarithm (base e).

3. What is the logarithm of 0?

Ans.It is undefined. No exponent of a positive base ever equals zero.

4. What is the logarithm of 10?

 Ans.log₁₀(10) = 1

5. What are key properties of logarithms?

Ans.

• $\log_{b}(mn) = \log_{b} m + \log_{b} n$

• $\log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n$
  
  

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