Logarithm and Its Properties

Definition

The logarithm of a number b to the base a is the power to which a must be raised to get b: logₐb = c ⇔ aᶜ = b.

• Condition for existence: a > 0, a ≠ 1, b > 0.
• Special notation: lg b = log₁₀b (common logarithm), ln b = logₑb (natural logarithm).
• log₂8 = 3, because 2³ = 8. logₐ1 = 0, logₐa = 1.

Basic Properties

Property	Formula
Product	logₐ(bc) = logₐb + logₐc
Quotient	logₐ(b/c) = logₐb − logₐc
Power	logₐ(bⁿ) = n · logₐb
Power in the base	the logarithm of b to the base aᵏ = (1/k) · logₐb
Basic identity	a^(logₐb) = b

Change of Base

• logₐb = lg b / lg a (changing to base 10).
• logₐb and the logarithm of a to the base b (b ≠ 1) are reciprocals of each other: their product is 1.
• logₐb · (the logarithm of c to the base b) = logₐc.

1. Check the condition for existence (a > 0, a ≠ 1, b > 0)
2. Reduce the expression to the same base
3. Apply the product, quotient and power properties
4. Calculate the result

Examples

• log₂32 + log₃9 = 5 + 2 = 7.
• log₆4 + log₆9 = log₆36 = 2.
• log₂3 · log₃8 = log₂8 = 3.
• If lg 2 = a, then lg 5 = lg(10/2) = 1 − a.
• 5^(log₅7 + 1) = 5^(log₅7) · 5 = 7 · 5 = 35.

Mistakes

• logₐ(b + c) ≠ logₐb + logₐc: there is no formula for the logarithm of a sum.
• logₐb · logₐc ≠ logₐ(bc): do not confuse a product with a sum.
• The logarithm of a negative number or of zero does not exist.