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CIE A-Level Maths Study Notes

2.2.1 Fundamentals of Logarithms and Indices

This comprehensive guide explores the intricate relationship between logarithms and indices are essential. It provides a deep dive into the core principles, laws, and applications of logarithms, offering students a thorough understanding of these fundamental mathematical concepts.

Understanding the Relationship Between Logarithms and Indices

Logarithms and indices, also known as exponents, are interconnected concepts in mathematics. A logarithm is essentially the inverse of exponentiation. In simpler terms, if a number is raised to a certain power to produce another number, the logarithm gives us the exponent that was used. This relationship is pivotal in understanding how logarithms operate and their applications in various mathematical problems.

The Concept of Logarithms

A logarithm answers the question: To what exponent must be raised a base number to obtain another number? For example, in the expression, bb is the base, aa is the number we are taking the logarithm of, and cc is the exponent to which bb must be raised to get aa. This forms the basis of understanding logarithmic functions.

Indices and Exponents

Indices, or exponents, are used to denote repeated multiplication of a number by itself. For instance, bnb^n means multiplying bb by itself nn times. The exponent nn can be any real number, leading to various types of exponentiation, including squares, cubes, and roots.

Laws of Logarithms

These laws, based on exponent properties, are essential for handling logarithmic expressions.

1. Product Law

  • Law: The logarithm of a product is the sum of the logarithms of the factors:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)
  • Example: Simplify log3(9)+log3(27)\log_3(9) + \log_3(27).
    • Solution: Apply the product law: log3(9×27)\log_3(9 \times 27). Calculate 9×27=2439 \times 27 = 243, and since 35=2433^5 = 243, log3(243)=5\log_3(243) = 5.
    • Conclusion: log3(9)+log3(27)=5\log_3(9) + \log_3(27) = 5.

2. Quotient Law

  • Law: The logarithm of a quotient is the difference between the logarithms of the numerator and denominator:
logb(xy)=logb(x)logb(y).\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y).
  • Example: Simplify log5(125)log5(25)\log_5(125) - \log_5(25).
    • Solution: Apply the quotient law: log5(12525)\log_5\left(\frac{125}{25}\right). Calculate 12525=5\frac{125}{25} = 5, and since 51=55^1 = 5, log5(5)=1\log_5(5) = 1.
    • Conclusion: log5(125)log5(25)=1\log_5(125) - \log_5(25) = 1.

3. Power Law

  • Law: The logarithm of a power is the product of the exponent and the logarithm of the base:
logb(xn)=nlogb(x).\log_b(x^n) = n\log_b(x).
  • Example: Simplify 2log4(16)2\log_4(16).
    • Solution: Apply the power law: log4(162)\log_4(16^2). Calculate 162=25616^2 = 256, and since 44=2564^4 = 256, log4(256)=4\log_4(256) = 4.
    • Conclusion: 2log4(16)=4 2\log_4(16) = 4.
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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