Properties of Logarithms

Because a logarithm is an exponent, the laws of exponents turn into three matching properties of logarithms: multiplying arguments adds their logs, dividing subtracts, and raising to a power multiplies by the log.

By the end you'll be able to expand a single logarithm into a sum/difference of simpler logs (and condense it back) using the product rule, quotient rule, and power rule.

Predict: with x = 8 and y = 4 on Product, what single power of 2 should log₂(8) + log₂(4) combine into? Combine the bars below and check against the check bar. (→ log₂ 32)

Each bar's length is a base-2 logarithm — how many times you'd multiply 2 by itself to reach the argument. Switch rules, drag the sliders, and watch the top bars visibly add, subtract, or scale into the same length as the bottom check bar — direct proof the rule holds.

log₂(8) + log₂(4) = 3.00 + 2.00 = 5.00 = log₂(32)

Top: the rule's bars — Bottom: the check bar, both measured in log₂ units
log₂(x) log₂(y) check: combined result

A logarithm turns multiplication into addition, division into subtraction, and exponentiation into ordinary multiplication — each rule drops the argument's operation down one level of difficulty.

Formal

For a base b > 0 (b ≠ 1) and positive arguments m and n:

  • Product rule: \(\log_b(mn) = \log_b(m) + \log_b(n)\)
  • Quotient rule: \(\log_b(m/n) = \log_b(m) - \log_b(n)\)
  • Power rule: \(\log_b(m^k) = k \cdot \log_b(m)\)

These fall straight out of the definition of a logarithm: \(\log_b(m)\) is just the exponent that produces m. Write \(m = b^p\) and \(n = b^q\); then \(mn = b^{p+q}\), so the exponent that produces \(mn\) is \(p+q\) — the product rule is exponent addition in disguise. Every rule needs a single common base and positive arguments; there is no rule for \(\log_b(m+n)\).

Procedural

Read left to right, these rules expand one logarithm into several simpler ones; read right to left, they condense several logs back into one. A companion tool is the change-of-base formula, \(\log_b(x) = \log_c(x)/\log_c(b)\) for any valid new base c — it lets a calculator that only knows log and ln evaluate a logarithm of any base.

Applied

The Richter scale, decibels, and pH are all logarithmic — each step up multiplies the underlying quantity. If a signal's power triples and then triples again, its logarithmic measure just adds \(\log_b(3)\) twice instead of forcing you to multiply the raw power out — the product rule is exactly why log scales make big multiplicative changes easy to add and compare.

Worked example

Expand \(\log_b(x^3 \cdot y / z)\).
1. Quotient rule splits the division: \(\log_b(x^3 y) - \log_b(z)\).
2. Product rule splits the multiplication: \(\log_b(x^3) + \log_b(y) - \log_b(z)\).
3. Power rule brings the exponent down: \(3\log_b(x) + \log_b(y) - \log_b(z)\).

Your turn

Expand \(\log_5(a^2 \cdot b / c)\).
1. Quotient rule: \(\log_5(a^2 b) - \log_5(c)\).
2. Product rule: \(\log_5(a^2) + \log_5(b) - \log_5(c)\).
3. Power rule: bring the 2 down. Finish it: ____

Reveal the answer

\(2\log_5(a) + \log_5(b) - \log_5(c)\) — the exponent 2 on a becomes a multiplying coefficient out front, while b and c (no exponent shown) stay as plain logs. Set the Power mode's p slider to 2 above and check the bars scale the same way.

More info — why these rules follow from the definition of a logarithm

A logarithm is defined as an inverse of exponentiation: \(\log_b(x)\) is the exponent you raise b to in order to get x. Exponent rules already tell you that same-base powers multiply by adding exponents (\(b^p \cdot b^q = b^{p+q}\)) and that a power raised to a power multiplies exponents (\((b^p)^k = b^{pk}\)). The product, quotient, and power rules for logarithms are just those exponent rules translated into "exponent language" — which is exactly what a logarithm names. See the Purplemath link in Dive deeper for more drill problems.

Check your understanding

Question 1 of 4

If \(\log_b(7) = 1.9\) and \(\log_b(3) = 1.1\), what is \(\log_b(21)\)?

Question 2 of 4

If \(\log_b(50) = 3.9\) and \(\log_b(2) = 0.7\), what is \(\log_b(25)\)?

Question 3 of 4

Which expression correctly expands \(\log_b(w^5)\)?

Question 4 of 4

Why does the product rule \(\log_b(mn) = \log_b(m) + \log_b(n)\) hold? (Think back to what a logarithm *is*.)

Recap

  • Product rule: \(\log_b(mn) = \log_b(m) + \log_b(n)\) — multiplying adds logs.
  • Quotient rule: \(\log_b(m/n) = \log_b(m) - \log_b(n)\) — dividing subtracts logs.
  • Power rule: \(\log_b(m^k) = k \cdot \log_b(m)\) — an exponent becomes a multiplying coefficient.
  • Every rule needs one common base and positive arguments; there is no rule for \(\log_b(m+n)\). The change-of-base formula \(\log_b(x) = \log_c(x)/\log_c(b)\) converts between bases.

Dive deeper

Sources

  • Properties of Logarithms