What are the main logarithm rules?

The main rules are the product rule log(xy) = log(x) + log(y), the quotient rule log(x/y) = log(x) - log(y), the power rule log(x^n) = n·log(x), and the change of base formula log_a(x) = log_b(x) / log_b(a). Each derives directly from corresponding exponent laws.

How do you expand a logarithmic expression?

To expand, apply the product rule to split multiplication into addition, the quotient rule to split division into subtraction, and the power rule to bring exponents out as coefficients. Work from the outermost operation inward until each term contains a single logarithm.

How do you condense logarithmic expressions?

To condense, reverse the expansion process. Convert coefficients into exponents using the power rule, combine sums into products using the product rule, and combine differences into quotients using the quotient rule. The result is a single logarithm.

Can you split the logarithm of a sum?

No. log(x + y) does not equal log(x) + log(y). The product, quotient, and power rules apply only to multiplication, division, and exponentiation inside the argument. There is no rule for sums or differences inside a logarithm.

What is the change of base formula used for?

The change of base formula log_a(x) = log_b(x) / log_b(a) converts a logarithm to any other base. It is commonly used to evaluate logarithms on a calculator by converting to base 10 or base e, since most calculators only have log and ln keys.

Logarithms Rules

The Change of Base Formula

Expanding Logarithmic Expressions

Condensing Logarithmic Expressions

Common Errors

Summary of Logarithm Rules

Transforming Products, Quotients, and Powers

The logarithm of a product, quotient, or power breaks apart in predictable ways. Each rule corresponds to an exponent law, inverted through the logarithmic lens. Mastering these rules enables expanding complex logarithmic expressions into simpler components and condensing sums or differences of logarithms into single expressions — both essential skills for solving equations.

Key Terms

The Rules

Product Rule (Logarithms)—

\log_a(xy) = \log_a(x) + \log_a(y)

Quotient Rule (Logarithms)—

\log_a(x/y) = \log_a(x) - \log_a(y)

Power Rule (Logarithms)—

\log_a(x^n) = n \cdot \log_a(x)

Change of Base Formula—

\log_a(x) = \log_b(x) / \log_b(a)

Supporting Concepts

Logarithm— each rule derives from an exponent law inverted through the definition

Common Logarithm— typical target base for change of base

Natural Logarithm— alternative target base

Formulas Used on This Page

Product Rule (Logarithms)—

\log_a(xy) = \log_a(x) + \log_a(y)

Quotient Rule (Logarithms)—

\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)

Power Rule (Logarithms)—

\log_a(x^n) = n \cdot \log_a(x)

Change of Base Formula—

\log_a(x) = \frac{\log_b(x)}{\log_b(a)}

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The Product Rule

The logarithm of a product equals the sum of the logarithms:

\log_a(xy) = \log_a(x) + \log_a(y)

Multiplication inside the argument becomes addition outside.

Derivation: Let

\log_a(x) = m

and

\log_a(y) = n

. Then

x = a^m

and

y = a^n

. The product

xy = a^m \cdot a^n = a^{m+n}

. Taking the logarithm:

\log_a(xy) = m + n = \log_a(x) + \log_a(y)

.

Example:

\log_2(8 \cdot 4) = \log_2(8) + \log_2(4) = 3 + 2 = 5

. Verification:

8 \cdot 4 = 32 = 2^5

.

The rule extends to any number of factors:

\log_a(xyz) = \log_a(x) + \log_a(y) + \log_a(z)

The Quotient Rule

The logarithm of a quotient equals the difference of the logarithms:

\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)

Division inside the argument becomes subtraction outside.

Derivation: Let

\log_a(x) = m

and

\log_a(y) = n

. Then

x = a^m

and

y = a^n

. The quotient

\frac{x}{y} = \frac{a^m}{a^n} = a^{m-n}

. Taking the logarithm:

\log_a(x/y) = m - n = \log_a(x) - \log_a(y)

.

Example:

\log_3(81/9) = \log_3(81) - \log_3(9) = 4 - 2 = 2

. Verification:

81/9 = 9 = 3^2

.

The quotient rule also yields:

\log_a(1/x) = \log_a(1) - \log_a(x) = 0 - \log_a(x) = -\log_a(x)

The Power Rule

The logarithm of a power equals the exponent times the logarithm:

\log_a(x^n) = n \cdot \log_a(x)

Exponents inside the argument become coefficients outside.

Derivation: Let

\log_a(x) = m

, so

x = a^m

. Then

x^n = (a^m)^n = a^{mn}

. Taking the logarithm:

\log_a(x^n) = mn = n \cdot \log_a(x)

.

Example:

\log_2(8^4) = 4 \cdot \log_2(8) = 4 \cdot 3 = 12

. Verification:

8^4 = 4096 = 2^{12}

.

The rule holds for any real exponent

n

, including fractions and negatives:

\log_a(\sqrt{x}) = \log_a(x^{1/2}) = \frac{1}{2}\log_a(x)

\log_a(x^{-3}) = -3\log_a(x)

The Change of Base Formula

A logarithm in one base can be expressed using logarithms in any other base:

\log_a(x) = \frac{\log_b(x)}{\log_b(a)}

This formula converts logarithms to a convenient base — typically

10

e

for calculator computation.

Derivation: Let

\log_a(x) = y

, so

a^y = x

. Taking

\log_b

of both sides:

\log_b(a^y) = \log_b(x)

. By the power rule:

y \cdot \log_b(a) = \log_b(x)

. Solving for

y

y = \frac{\log_b(x)}{\log_b(a)}

.

Common applications:

\log_2(10) = \frac{\log(10)}{\log(2)} = \frac{1}{0.301} \approx 3.322

\log_5(7) = \frac{\ln(7)}{\ln(5)} = \frac{1.946}{1.609} \approx 1.209

A useful special case:

\log_a(b) = \frac{1}{\log_b(a)}

. The logarithms are reciprocals when arguments and bases swap.

The four rules above all follow the same shape: name the logarithms, convert to exponential form, apply the matching exponent law, then read the result back as a logarithm. The table below sets the four derivations side by side so the shared pattern is visible at a glance.

Rule	Setup	Exponent law applied	Read off as a logarithm
Product	let logₐ(x) = m, logₐ(y) = n, so x = aᵐ, y = aⁿ	aᵐ · aⁿ = aᵐ⁺ⁿ	logₐ(xy) = m + n = logₐ(x) + logₐ(y)
Quotient	same setup: x = aᵐ, y = aⁿ	aᵐ ⁄ aⁿ = aᵐ⁻ⁿ	logₐ(x⁄y) = m − n = logₐ(x) − logₐ(y)
Power	let logₐ(x) = m, so x = aᵐ	(aᵐ)ⁿ = aᵐⁿ	logₐ(xⁿ) = mn = n · logₐ(x)
Change of base	let logₐ(x) = y, so aʸ = x	take log_b of both sides; apply the power rule	y · log_b(a) = log_b(x), so logₐ(x) = log_b(x) ⁄ log_b(a)

Expanding Logarithmic Expressions

A single logarithm with a compound argument can be written as a sum or difference of simpler logarithms, with exponents pulled out as coefficients. The product rule splits multiplication, the quotient rule splits division, and the power rule extracts exponents.

Example: Expand

\log_2\left(\frac{x^3 y}{z^2}\right)

\log_2\left(\frac{x^3 y}{z^2}\right) = \log_2(x^3 y) - \log_2(z^2)

= \log_2(x^3) + \log_2(y) - \log_2(z^2)

= 3\log_2(x) + \log_2(y) - 2\log_2(z)

Example: Expand

\ln\left(\sqrt{\frac{a}{b^3}}\right)

\ln\left(\sqrt{\frac{a}{b^3}}\right) = \ln\left(\frac{a}{b^3}\right)^{1/2} = \frac{1}{2}\ln\left(\frac{a}{b^3}\right)

= \frac{1}{2}\left(\ln(a) - \ln(b^3)\right) = \frac{1}{2}\ln(a) - \frac{3}{2}\ln(b)

Condensing Logarithmic Expressions

Multiple logarithms can be combined into a single logarithm by reversing the expansion process. Coefficients become exponents through the power rule, sums become products through the product rule, and differences become quotients through the quotient rule.

Example: Condense

2\log(x) + \log(y) - 3\log(z)

2\log(x) + \log(y) - 3\log(z) = \log(x^2) + \log(y) - \log(z^3)

= \log(x^2 y) - \log(z^3) = \log\left(\frac{x^2 y}{z^3}\right)

Example: Condense

\frac{1}{2}\ln(a+1) - \ln(b) - \ln(c)

\frac{1}{2}\ln(a+1) - \ln(b) - \ln(c) = \ln(a+1)^{1/2} - \ln(b) - \ln(c)

= \ln\sqrt{a+1} - \ln(bc) = \ln\left(\frac{\sqrt{a+1}}{bc}\right)

Condensing is essential for solving equations where multiple logarithms must combine before applying the one-to-one property.

Common Errors

Several incorrect "rules" persistently appear. None of the following equalities hold:

\log_a(x + y) \neq \log_a(x) + \log_a(y)

The logarithm of a sum does not split. There is no simplification for

\log_a(x + y)

\log_a(x - y) \neq \log_a(x) - \log_a(y)

The logarithm of a difference does not split either.

\log_a(x) \cdot \log_a(y) \neq \log_a(xy)

Multiplying logarithms is not the same as the logarithm of a product.

\frac{\log_a(x)}{\log_a(y)} \neq \log_a\left(\frac{x}{y}\right)

Dividing logarithms is not the same as the logarithm of a quotient. However, note that

\frac{\log_a(x)}{\log_a(y)} = \log_y(x)

by the change of base formula.

(\log_a(x))^n \neq n\log_a(x)

Raising a logarithm to a power is not the same as multiplying by that power. The power rule applies to exponents inside the argument, not outside.

Incorrect "rule"	What is actually true
logₐ(x + y) ≠ logₐ(x) + logₐ(y)	sums inside a logarithm have no expansion — leave logₐ(x + y) as written
logₐ(x − y) ≠ logₐ(x) − logₐ(y)	differences inside a logarithm have no expansion either
logₐ(x) · logₐ(y) ≠ logₐ(xy)	the product rule equates logₐ(xy) with the sum logₐ(x) + logₐ(y), not the product
logₐ(x) ⁄ logₐ(y) ≠ logₐ(x⁄y)	this ratio equals log_y(x) by change of base — a different logarithm entirely
(logₐ(x))ⁿ ≠ n · logₐ(x)	the power rule applies to exponents inside the argument; an exponent on the whole logarithm does not move

Summary of Logarithm Rules

The four rules together form a single transformation system: each one converts an operation inside the argument into a different operation on the outside. The table below collects the rules in one place — identity, what gets transformed, and a quick numerical example — useful as a reference card while expanding or condensing expressions.

Rule	Identity	What it transforms	Example
Product	logₐ(x·y) = logₐ(x) + logₐ(y)	multiplication inside ↔ addition outside	log₂(8·4) = 3 + 2 = 5
Quotient	logₐ(x⁄y) = logₐ(x) − logₐ(y)	division inside ↔ subtraction outside	log₃(81⁄9) = 4 − 2 = 2
Power	logₐ(xⁿ) = n · logₐ(x)	exponent inside ↔ coefficient outside	log₂(8³) = 3 · 3 = 9
Change of base	logₐ(x) = log_b(x) ⁄ log_b(a)	rewrites a logarithm in any new base b	log₂(7) = log(7)⁄log(2) ≈ 2.807