What is log₆ 4 + log₆ 9? Neither logarithm comes out as a whole number. Their sum does, though: 2. Why? Because 4 · 9 = 36 = 6². The sum of the logarithms equals the logarithm of the product.
A logarithm is an exponent, so it follows the same rules as exponents. From the rule "when you multiply powers, you add the exponents" we get the law "the log of a product is the sum of the logs".
For positive numbers x, y and an allowed base a (a > 0, a ≠ 1):
Law
Formula
Example
Log of a product
logₐ (x · y) = logₐ x + logₐ y
log₂ (4 · 8) = 2 + 3 = 5
Log of a quotient
logₐ (x / y) = logₐ x − logₐ y
log₃ (81 / 9) = 4 − 2 = 2
Log of a power
logₐ (xᵏ) = k · logₐ x
log₂ (8⁵) = 5 · 3 = 15
In words: product → sum, quotient → difference, exponent → multiplier in front of the logarithm.
Why the laws work
Let logₐ x = r and logₐ y = s. From the definition, x = aʳ and y = aˢ. Then
x · y = aʳ · aˢ = a^(r + s),
and so logₐ (x · y) = r + s = logₐ x + logₐ y. The laws for a quotient and a power follow in the same way from the rules aʳ / aˢ = a^(r − s) and (aʳ)ᵏ = a^(k·r).
Calculations without a calculator
Worked example 1
log₆ 4 + log₆ 9 = log₆ (4 · 9) = log₆ 36 = 2.
Worked example 2
log₂ 40 − log₂ 5 = log₂ (40 / 5) = log₂ 8 = 3.
Worked example 3
2 · log₃ 6 − log₃ 4
Move the 2 into the exponent: 2 · log₃ 6 = log₃ 6² = log₃ 36.
log₃ 36 − log₃ 4 = log₃ (36 / 4) = log₃ 9.
log₃ 9 = 2.
Expanding and combining logarithms
In algebra we work with variables. For positive x, y, z:
Expanding
logₐ (x² · y / z³) = 2 · logₐ x + logₐ y − 3 · logₐ z.
The numerator gives plus signs, the denominator a minus sign, and the exponents move in front of the logarithm. A square root is a power with the exponent ½: logₐ √z = ½ · logₐ z.
Combining into a single logarithm
3 · logₐ x + logₐ y − 2 · logₐ z = logₐ (x³ · y / z²).
The method works the other way round: move the numbers in front of the logarithms into the exponents, put the logarithms with a plus sign in the numerator and those with a minus sign in the denominator.
Change of base
A calculator only knows the common logarithm (log, base 10). We convert any other base with the formula
logₐ b = log b / log a
Worked example 4
log₂ 3 · log₃ 8
Write both logarithms as common logarithms: (log 3 / log 2) · (log 8 / log 3).
log 3 cancels: log 8 / log 2 = log₂ 8.
log₂ 8 = 3.
💡 The formula also shows that, for example, log₂ 5 · log₅ 2 = 1.
Common mistakes
logₐ (x + y) = logₐ x + logₐ y. No! The law works for a product, not for a sum. For example, log₂ (4 + 4) = 3, but log₂ 4 + log₂ 4 = 4.
logₐ x · logₐ y = logₐ (x · y). No, a product of logarithms does not simplify like this.
logₐ x / logₐ y = logₐ (x / y). No, a quotient of logarithms is a change of base: it is log base y of x.
(logₐ x)² = 2 · logₐ x. No, only an exponent on the argument can be moved: logₐ (x²) = 2 · logₐ x.
Summary
logₐ (x · y) = logₐ x + logₐ y.
logₐ (x / y) = logₐ x − logₐ y.
logₐ (xᵏ) = k · logₐ x.
logₐ b = log b / log a (change of base).
The laws do not work for the log of a sum or for a product of logs.