Rules for Calculating Logarithms

INTRODUCTION TO CALCULATION RULES

Logarithms transform multiplication into addition, division into subtraction, and exponentiation (raising to a power) into multiplication. Because of these properties, logarithm rules are key for efficiently simplifying mathematical expressions. We use them when solving logarithmic equations, expressing exponential relationships, and breaking down complex expressions.

Logarithm rules are based on the definition of a logarithm and the properties of exponential expressions.

RULES OF LOGARITHMIC CALCULATION

1. PRODUCT RULE:
   logₐ(x * y) = logₐ(x) + logₐ(y)
   The logarithm of a product is the sum of the logarithms of the factors.
2. QUOTIENT RULE:
   logₐ(x / y) = logₐ(x) – logₐ(y)
   The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator.
3. POWER RULE:
   logₐ(xⁿ) = n * logₐ(x)
   The logarithm of a power is the exponent times the logarithm of the base of the power.
4. ROOT RULE (SPECIAL CASE OF THE POWER RULE):
   logₐ(ⁿ√x) = logₐ(x^(1/n)) = (1/n) * logₐ(x)
   (The Slovenian text uses √x implying square root, so logₐ(√x) = (1/2) * logₐ(x) is a specific instance.)
   A root is a power with a rational exponent. For a square root: logₐ(√x) = (1/2) * logₐ(x).
5. LOGARITHM OF THE BASE:
   logₐ(a) = 1
   The logarithm of a number equal to its base is always 1 (since a¹ = a).
6. LOGARITHM OF ONE:
   logₐ(1) = 0
   Since a⁰ = 1 for any valid base a (a > 0, a ≠ 1), it follows that logₐ(1) = 0.
7. CHANGE OF BASE FORMULA (CONVERSION BETWEEN BASES):
   logₐ(x) = log_b(x) / log_b(a)
   The logarithm to any base can be expressed using logarithms to another base.

EXAMPLES OF USING THE RULES

• log₂(8 * 4) = log₂(8) + log₂(4) = 3 + 2 = 5
• log₁₀(100 / 10) = log₁₀(100) – log₁₀(10) = 2 – 1 = 1
• log₃(27²) = 2 * log₃(27) = 2 * 3 = 6
• log₄(√16) = (1/2) * log₄(16) = (1/2) * 2 = 1 (using √16 = 16^(1/2))

CONCLUSION

The rules for calculating logarithms allow for the transformation and simplification of logarithmic expressions. By using these laws, we can quickly switch between multiplication, division, and exponentiation in logarithmic form. Each rule stems from the basic characteristics of exponential functions, which gives logarithms an important role in bridging the gap between linear and exponential structures in mathematics.