Laws of Logarithms: Formulas, Rules and Examples

Laws of Logarithms provide rules for simplifying products, quotients and powers inside logarithmic expressions. The main rules include the product law, quotient law, power law and change-of-base formula. This page explains their conditions, standard log formulas, common simplification methods and short numerical examples for quantitative aptitude questions.

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What Are the Laws of Logarithms?

Laws of logarithms are algebraic rules used to combine or simplify logarithmic expressions. For base a, the base must satisfy a > 0 and a ≠ 1, while every logarithm argument must be positive.

For positive numbers M and N, the basic laws are logₐ(MN) = logₐM + logₐN, logₐ(M/N) = logₐM − logₐN and logₐ(Mⁿ) = n logₐM. The change-of-base rule is logₐM = log_bM / log_ba, where b is any valid base. For example, log₂(8 × 4) = log₂8 + log₂4 = 3 + 2 = 5.

Laws of Logarithms Formula & Tricks

Important Formulas

Product law
logₐ(MN) = logₐM + logₐN

The logarithm of a product equals the sum of the logarithms of its positive factors.

Quotient law
logₐ(M/N) = logₐM − logₐN

The logarithm of a quotient equals the difference of the logarithms of the numerator and denominator.

Power law
logₐ(Mⁿ) = n logₐM

An exponent inside a logarithm becomes a multiplier outside it, provided M > 0.

Change of base
logₐM = log_bM / log_ba

A logarithm can be changed from base a to any base b that is positive and not equal to 1.

Reciprocal relation
logₐb = 1 / log_ba

Interchanging the base and argument gives reciprocal logarithmic values.

Logarithm of a base power
logₐ(aˣ) = x

A logarithm and an exponential with the same base cancel each other.

Quick Tricks

Convert products and quotients before calculating

Use the product law to split multiplication and the quotient law to split division. This often changes a difficult expression into known logarithms.

Example: log₂(32/4) = log₂32 − log₂4 = 5 − 2 = 3.
Move powers outside the logarithm

Apply the power law before evaluating. A square, cube or other exponent becomes a coefficient.

Example: log₃(81²) = 2 log₃81 = 2 × 4 = 8.
Use reciprocal values for reversed bases

When the base and argument are interchanged, use logₐb = 1/log_ba instead of recalculating.

Example: Since log₂8 = 3, log₈2 = 1/3.

Laws of Logarithms Concepts

Product, Quotient and Power Laws

The product law changes multiplication into addition, the quotient law changes division into subtraction, and the power law changes an exponent into a multiplier.

For M > 0 and N > 0, logₐ(MN) = logₐM + logₐN and logₐ(M/N) = logₐM − logₐN. Also, logₐ(Mⁿ) = n logₐM. These rules work for any valid logarithm base. Example: log₅(25 × 125) = log₅25 + log₅125 = 2 + 3 = 5.

Example: log₇(49³/7²) = 3 log₇49 − 2 log₇7 = 3 × 2 − 2 × 1 = 4.

Change-of-Base Formula

The change-of-base formula expresses logₐM using logarithms with another base: logₐM = log_bM/log_ba.

The new base b must satisfy b > 0 and b ≠ 1. Common logarithms use base 10, so log₂8 = log 8/log 2 = 3. The formula also allows conversion to natural logarithms: logₐM = ln M/ln a.

Example: log₄64 = log 64/log 4 = 6/2 = 3.

Standard Logarithm Values and Identities

The standard identities are logₐ1 = 0, logₐa = 1, logₐ(aˣ) = x and a^(logₐM) = M.

The identity logₐ1 = 0 follows because a⁰ = 1. Similarly, logₐa = 1 because a¹ = a. The expressions a^(logₐM) and logₐ(aˣ) cancel only when the bases match and the logarithm argument is valid.

Example: log₉1 = 0, log₉9 = 1 and 9^(log₉5) = 5.

Conditions and Invalid Logarithmic Expressions

A logarithm logₐM is defined in the real number system only when a > 0, a ≠ 1 and M > 0.

The argument cannot be zero or negative. Therefore, log₂0 and log₂(−8) are undefined in real numbers. In log₃(x − 2), the condition is x − 2 > 0, so x > 2. The base condition applies separately to every logarithm.

Example: For log₅(2x + 1), the argument condition is 2x + 1 > 0, giving x > −1/2.

Combining Logarithms into One Expression

Separate logarithms can be combined by reversing the product, quotient and power laws.

The sum logₐM + logₐN becomes logₐ(MN), while the difference logₐM − logₐN becomes logₐ(M/N). A coefficient can be placed inside as an exponent: k logₐM = logₐ(Mᵏ), for M > 0.

Example: 2 log₃5 + log₃7 − log₃2 = log₃(5² × 7/2) = log₃(175/2).

Laws of Logarithms Video Lessons

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Logarithms: Definition and Fundamental Laws

Understand the definition of logarithms and learn their fundamental laws, including the rules for simplifying logarithmic expressions in quantitative aptitude.

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Practice Laws of Logarithms Questions

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1If 5^(x+1) = 1 , what is the value of x?→ 2Find the value of { ( 1/3 )⁻² - ( 1/2 )⁻³ } ÷ ( 1/4 )⁻² + { ( (-3)/2 )⁴ × ( 5/3 )⁴ } .→ 3Simplify: (2³)² ÷ 2⁴→ 4Simplify: √(5 + √25 + 5² - 3² - 10) - √4→ 5Simplify the expression ((49)^n × 7)/(7^(2n)) . If n is a positive integer, what is the result in terms of powers of 7 ?→ 6Simplify: (√98 + √200 - 2√8)²→ 7The value of the expression 0.125 × 4⁴ × 2⁻⁵ × 0.25² × 8² is ______.→ 8Evaluate: √(250 + √(19 + √289)) + √(620 + √(11 + √196))→ 9Find the value of m which satisfies (19/10)⁷ × (10/19)⁶ × (19/10)¹⁰ = (10/19)^(6m + 18)→ 10Simplify ((32)6)5 and express it as a power of 9. What is the exponent of 9?→

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Quick Revision Notes

Laws of Logarithms: Quick Revision

Use these rules to simplify and evaluate logarithmic expressions.

  • logₐ(MN) = logₐM + logₐN.
  • logₐ(M/N) = logₐM − logₐN.
  • logₐ(Mⁿ) = n logₐM.
  • logₐM = log_bM/log_ba.
  • logₐ1 = 0 and logₐa = 1.
  • logₐ(aˣ) = x and a^(logₐM) = M.
  • For real logarithms, a > 0, a ≠ 1 and the argument is greater than zero.
  • logₐb and log_ba are reciprocals.

Laws of Logarithms FAQs

What is the product law of logarithms?

The product law is logₐ(MN) = logₐM + logₐN, where M and N are positive. For example, log₂(8 × 4) = 3 + 2 = 5.

What is the quotient law of logarithms?

The quotient law is logₐ(M/N) = logₐM − logₐN. Thus, log₃(81/9) = 4 − 2 = 2.

How is a power inside a logarithm simplified?

Use logₐ(Mⁿ) = n logₐM. For example, log₂(16³) = 3 log₂16 = 3 × 4 = 12.

What is the change-of-base formula for logarithms?

The formula is logₐM = log_bM/log_ba. Therefore, log₄64 = log 64/log 4 = 6/2 = 3.

What is the value of logₐ1?

logₐ1 = 0 because a⁰ = 1, for every valid base a.

What is the value of logₐa?

logₐa = 1 because a¹ = a, provided a > 0 and a ≠ 1.

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