Logarithm Formulas, Rules, Examples & Log Table

Logarithm Formulas, Rules, Examples & Log Table

Logarithm Formulas, Rules, Examples & Log Table

Logarithms play a significant role in mathematics and real-world applications, making them an essential topic to understand. In this article, we will explore logarithm formulas, rules, properties, and provide solved examples to clarify concepts.

Table of Contents

  1. What is a Logarithm?
  2. Key Logarithm Concepts
  3. Logarithm Rules and Properties
  4. Finding the Characteristic and Mantissa
  5. Applications of Logarithms
  6. Solved Examples
  7. FAQs on Logarithms

What is a Logarithm?

In mathematics, a logarithm is the inverse function of exponentiation. The logarithm of a number to a given base is the power to which the base must be raised to produce the number.

Example:

If ax=Na^x = N, then x=log⁡aNx = \log_a N. This is read as “log N to the base a.”Points to Note:
  • NN: A positive number.
  • aa: A positive number other than 1.

Common Terminologies:

  • Common Logarithms: Base 10 logarithms.
  • Natural Logarithms: Base ee logarithms.
  • If no base is mentioned, the logarithm is assumed to have base 10.

Key Logarithm Concepts

  1. Characteristic and Mantissa
    • The integral part of a common logarithm is called the characteristic.
    • The decimal part is called the mantissa.

Example:

For log⁡5123=3.709\log 5123 = 3.709:
  • Characteristic: 3.
  • Mantissa: 0.709.
Key Rules for the Characteristic:
  • For numbers >1> 1: The characteristic is one less than the number of digits to the left of the decimal point.
  • For numbers <1< 1: The characteristic is negative and equal to −1-1 times the number of zeros between the decimal point and the first significant digit.

Example:

  • log⁡3456.25\log 3456.25: Characteristic = 3.
  • log⁡0.0045\log 0.0045: Characteristic = -3.
.

Logarithm Rules and Properties

Formulas:

  1. log⁡aa=1\log_a a = 1
  2. log⁡a1=0\log_a 1 = 0
  3. log⁡a(m⋅n)=log⁡am+log⁡an\log_a (m \cdot n) = \log_a m + \log_a n
  4. log⁡a(mn)=log⁡am−log⁡an\log_a \left(\frac{m}{n}\right) = \log_a m – \log_a n
  5. log⁡a(1/m)=−log⁡am\log_a (1/m) = -\log_a m
  6. log⁡a(mp)=p⋅log⁡am\log_a (m^p) = p \cdot \log_a m
  7. log⁡am=log⁡bmlog⁡ba\log_a m = \frac{\log_b m}{\log_b a}
  8. alog⁡bc=clog⁡baa^{\log_b c} = c^{\log_b a}
  9. log⁡ab=1log⁡ba\log_a b = \frac{1}{\log_b a}
Note: The base aa of a logarithm cannot be 1. 

Finding the Characteristic and Mantissa

To calculate the characteristic and mantissa:
  1. Convert the number to standard form. Example: Convert 3071 to 3.071×1033.071 \times 10^3.
  2. Use the formula:log⁡n=log⁡t+p,\log n = \log t + p,where:
    • n=t⋅10pn = t \cdot 10^p
    • pp: Characteristic
    • log⁡t\log t: Mantissa
 

Applications of Logarithms

  1. Finding the Number of Digits in Large Numbers
    • The number of digits in NN: ⌊log⁡10N⌋+1\lfloor \log_{10} N \rfloor + 1.

Example:

48124812: log⁡104812≈3.6826\log_{10} 4812 \approx 3.6826 Number of digits = ⌊3.6826⌋+1=4\lfloor 3.6826 \rfloor + 1 = 4.
  1. Scientific Calculations Logarithms simplify calculations involving multiplication or division of large numbers.
  2. Applications in Various Fields
    • Data science, physics, and engineering.
    • Modeling exponential growth or decay (e.g., population, radioactive decay).
 

Solved Examples

Example 1:

Find the number of digits in 22562^{256}:Solution:log⁡10(2256)=256⋅log⁡102=256⋅0.30103=77.056.\log_{10} (2^{256}) = 256 \cdot \log_{10} 2 = 256 \cdot 0.30103 = 77.056.Number of digits = ⌊77.056⌋+1=78\lfloor 77.056 \rfloor + 1 = 78.

Example 2:

Evaluate log⁡51024\log_5 1024:Solution:log⁡51024=log⁡101024log⁡105=10⋅log⁡1021−log⁡102.\log_5 1024 = \frac{\log_{10} 1024}{\log_{10} 5} = \frac{10 \cdot \log_{10} 2}{1 – \log_{10} 2}. 10⋅0.301031−0.30103=3.01030.69897≈4.3062.\frac{10 \cdot 0.30103}{1 – 0.30103} = \frac{3.0103}{0.69897} \approx 4.3062.

Example 3:

Solve for xx in log⁡102+log⁡10(6x+1)=log⁡10(x+5)+1\log_{10} 2 + \log_{10} (6x + 1) = \log_{10} (x + 5) + 1:Solution:log⁡10[2(6x+1)]=log⁡10[10(x+5)].\log_{10} [2(6x + 1)] = \log_{10} [10(x + 5)].Equating:2(6x+1)=10(x+5)  ⟹  x=24.2(6x + 1) = 10(x + 5) \implies x = 24.

FAQs on Logarithms

  1. What is the base of a common logarithm? The base of a common logarithm is 10.
  2. Can the base of a logarithm be negative? No, the base must always be a positive number greater than 0, excluding 1.
  3. What is the logarithm of 1 to any base? log⁡a1=0\log_a 1 = 0 for any base a>0a > 0.

Conclusion

Logarithms are a crucial mathematical concept with extensive applications in various fields like engineering, science, and competitive exams. Understanding logarithmic formulas, rules, and their practical usage simplifies complex calculations and problem-solving. By mastering these concepts and using log tables effectively, you can tackle logarithmic problems with confidence and precision.
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