Logarithms Chapter Notes | Mathematics Class 9 ICSE PDF Download

Introduction

Imagine a world where complex calculations become as simple as adding or subtracting numbers! That's the magic of logarithms, a powerful mathematical tool that transforms multiplication into addition, division into subtraction, and powers into simpler forms. In this chapter, we'll dive into the fascinating world of logarithms, exploring how they help us express relationships between numbers in a new way, making calculations easier and more intuitive. Whether you're solving exponential equations or simplifying large numbers, logarithms are like a shortcut that unlocks the secrets of numbers. Let's embark on this journey to understand logarithms step by step!

​Logarithms 

• Logarithms simplify complex calculations involving exponents.
• They express the relationship between a base, an exponent, and a result in a different form.
• If a^b = c, then log_ac = b, where a ≠ 1.
• This is read as "logarithm of c to the base a is b."
• The exponential form (a^b = c) and logarithmic form (log_ac = b) are interchangeable.

Interchanging (Logarithmic form vis-à-vis exponential form)

• Exponential form is a^b = c, where a is the base, b is the exponent, and c is the result.
• Logarithmic form is log_ac = b, showing the same relationship differently.
• Logarithm of 1 to any base is always 0 (since a⁰ = 1).
• Logarithm of a number to its own base is always 1 (since a¹ = a).

Laws of Logarithm with Use

Logarithm laws help simplify expressions involving products, quotients, and powers.

First Law (Product Law)

• The logarithm of a product is the sum of the logarithms of its factors with the same base.
• Formula: log_a(m × n) = log_am + log_an.
• This applies to multiple factors: log_a(m × n × p) = log_am + log_an + log_ap.
• Note: log_a(m + n) ≠ log_am + log_an.

Second Law (Quotient Law)

• The logarithm of a fraction is the logarithm of the numerator minus the logarithm of the denominator, both with the same base.
• Formula: log_a(m/n) = log_am - log_an.
• Note: log_am / log_an ≠ log_am - log_an, and log_a(m - n) ≠ log_am - log_an.

Third Law (Power Law)

• The logarithm of a number raised to a power is the power times the logarithm of the number with the same base.
• Formula: log_a(mⁿ) = n log_am.
• Corollary: For roots, log_a(m^1/n) = (1/n) log_am.

Additional Notes

• Logarithms to base 10 are called common logarithms.
• If no base is specified, the base is assumed to be 10 (e.g., log 8 = log₁₀8).
• log₁₀1 = 0, log₁₀10 = 1, log₁₀100 = 2, log₁₀1000 = 3, etc.

Expansion of Expressions with the Help of Laws of Logarithms

• Logarithm laws are used to expand or combine logarithmic expressions.
• Product law breaks products into sums, quotient law handles fractions, and power law handles exponents.
• Conversely, logarithmic sums or differences can be combined into a single logarithm.

More About Logarithms

Logarithms have additional properties that relate bases and arguments.

Property 1: Reciprocal Relationship

• If log_ba = x, then log_ab = 1/x.
• This shows that the logarithm of a to base b is the reciprocal of the logarithm of b to base a.

Property 2: Product of Reciprocal Logarithms

• log_ba × log_ab = 1.
• This follows from the reciprocal relationship.

Property 3: Logarithm of a Base to Itself

• log_aa = 1, so log_a(a^x) = x.
• This is because raising a base to a power x gives a^x, and the logarithm reverses this.

Property 4: Change of Base

• log_ba = (log_xa) / (log_xb), where x is any positive number.
• This allows logarithms with different bases to be expressed in terms of a common base.