Logarithm Rules 1

INTRODUCTION TO LOGARITHMS AND THEIR RULES

Logarithms are mathematical tools used to solve equations where the unknown is in the exponent. The fundamental definition of a logarithm states that log_b(x) = y if and only if b^y = x, where 'b' is a positive number not equal to 1, and 'x' is a positive real number. To facilitate working with logarithms, there are established rules derived from the laws of exponentiation that allow logarithmic expressions to be simplified and reformulated.

BASIC RULES FOR LOGARITHMS

PRODUCT RULE

The logarithm of the product of two numbers is equal to the sum of the logarithms of the individual factors:
log_b(x * y) = log_b(x) + log_b(y)
Example:
log₂(8) = log₂(4 * 2) = log₂(4) + log₂(2) = 2 + 1 = 3

QUOTIENT RULE

The logarithm of the quotient of two numbers is equal to the difference of their logarithms:
log_b(x / y) = log_b(x) - log_b(y)
Example:
log₂(8 / 4) = log₂(8) - log₂(4) = 3 - 2 = 1

POWER RULE

The logarithm of a power is equal to the product of the exponent and the logarithm of the base of the power:
log_b(xⁿ) = n * log_b(x)
Example:
log₂(8) = log₂(2³) = 3 * log₂(2) = 3 * 1 = 3

CHANGE OF BASE RULE

The logarithm to base 'b' can be expressed using the logarithm of any other base 'c':
log_b(x) = log_c(x) / log_c(b)
Example:
log₂(8) = log₁₀(8) / log₁₀(2) ≈ 0.903 / 0.301 ≈ 3

LOGARITHM OF UNITY AND LOGARITHM OF THE BASE

• log_b(1) = 0, since b⁰ = 1
• log_b(b) = 1, since b¹ = b
  Examples:
  log₂(1) = 0
  log₂(2) = 1

RELATIONSHIP WITH EXPONENTIAL FUNCTIONS

The logarithm is the inverse function of the exponential function. This means:
If b^y = x, then y = log_b(x).

Example for solving an equation:
If we have 2^x = 8, then by taking the logarithm of both sides (to base 2):
x = log₂(8) = 3

CONCLUSION

The rules for logarithms are fundamental for understanding and working with logarithms. They enable us to:

• Simplify complex expressions
• Solve logarithmic equations
• Convert between different logarithm bases
• Work accurately in algebra and analysis

Knowledge of these rules provides us with a more powerful tool for mastering exponential expressions and a broader mathematical understanding.