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Logarithm Rules - Product, Quotient, and Power

TL;DR

Three logarithm rules turn complicated arithmetic into simple arithmetic: the product rule (log⁡b(xy)=log⁡bx+log⁡by) turns multiplication into addition; the quotient rule turns division into subtraction; the power rule turns exponents into multiplication.

What Are Logarithm Rules?

A logarithm answers the question "to what power must I raise the base to get this number?" If by=xb^y = x, then log⁡b(x)=y. For example, log⁡2(8)=3 because 2^3 = 8.

The logarithm rules are a small set of identities that let you rewrite log expressions in simpler forms. Three rules carry almost all of the work:

The Three Core Logarithm Rules

1. The Product Rule

log⁡b(xy)=log⁡b(x)+log⁡b(y)

In words: The log of a product equals the sum of the logs.

Example: log⁡2(8×4)=log⁡2(8)+log⁡2(4)=3+2=5. Check: log⁡2(32)=5 because 2^5 = 32. ✓

Why it works: Let m=log⁡b(x) and n=log⁡b(y). Then x=b^m and y=b^n. So xy=b^m·b^n. Taking the log of both sides: log⁡b(xy)=m+n=log⁡b(x)+log⁡b(y).

2. The Quotient Rule

log⁡b!(xy)=log⁡b(x)−log⁡b(y)

In words: The log of a quotient equals the log of the numerator minus the log of the denominator.

Example: log⁡2!(16/4)=log⁡2(16)−log⁡2(4)=4−2=2. Check: log⁡2(4)=2. ✓

Why it works: If x=b^m and y=b^n, then xy=b^{m−n}. Taking log⁡b of both sides gives: log⁡b!(x/y)=m−n.

3. The Power Rule

log⁡b(xn)=nlog⁡b(x)

In words: The log of a power equals the exponent times the log of the base.

Example: log⁡2(8^3)=3log⁡2(8)=3·3=9. Check: 8^3=512 and log⁡2(512)=9. ✓

Why it works: If x=b^m, then x^n=(b^m)^n=b^{mn}. Taking log⁡b gives: log⁡b(x^n)=mn=nlog⁡b(x).

How Do You Prove the Three Logarithm Rules?

Each of the three core rules has a short proof — and the proofs all rest on the same idea: a logarithm is just the inverse of an exponential.

Proof of the Product Rule

Let m=log⁡bx and n=log⁡by. By the definition of logarithm: b^m=x and b^n=y. Multiply: xy=b^m·b^n. Taking log⁡b of both sides gives: log⁡b(xy)=m+n=log⁡bx+log⁡by.

Proof of the Quotient Rule

Let m=log⁡bx and n=log⁡by, so x=b^m and y=b^n. Divide: xy=b^m/b^n, so taking log⁡b gives: log⁡b!(xy)=m−n.

Proof of the Power Rule

Let m=log⁡bx. Raise to the power n: x^n=(b^m)^n=b^{mn}. Taking log⁡b gives: log⁡b(x^n)=mn=nlog⁡bx.

Two Supporting Identities

Alongside the three core rules, two additional identities are worth memorising:

• log⁡b(1)=0
• log⁡b(b)=1

These follow directly from the definition.

The Change-of-Base Formula

To compute a log in any other base, use:

log⁡b(x)=log⁡c(x)/log⁡c(b)

where c is any base you can actually compute.

Example: Compute log⁡2(50).

log⁡2(50)=log⁡10(50)/log⁡10(2)≈1.699/0.301≈5.64.

Where Logarithms Appear in the Real World

• Richter scale. A magnitude-7 earthquake releases about 10^10 times more energy than a magnitude-6. The scale is log₁₀.
• Decibels. A jet engine (140 dB) is 10^(14/10)≈25,000,000 times more intense than ordinary conversation (60 dB).
• pH scale. A pH-3 acid is 10 times more acidic than pH-4.
• Stellar magnitudes. Each magnitude step is approximately 2.512 times the brightness.
• Compound interest and growth. The time required to double your money at rate r is t=log(2)/log(1+r).
• Information theory. Claude Shannon's definition of information is log₂-based.

A Worked Example — Wrong Path First

Simplify log⁡2(8·4).

The intuitive (wrong) approach. A student may apply the product rule incorrectly.

The correct method: log⁡2(8·4)=log⁡2(8)+log⁡2(4)=3+2=5.

Common Mistakes with Logarithm Rules

Mistake 1: Splitting the log of a sum

This is false. log(x+y) cannot be simplified further.

Mistake 2: Splitting the log of a product as a product of logs

log⁡2(8·4)=log⁡2(8)+log⁡2(4) not log⁡2(8)·log⁡2(4).

Mistake 3: Power rule with the wrong exponent location

(log⁡x)² and log(x²) are different.

The Mathematicians Who Shaped Logarithms

John Napier (1550–1617, Scotland) — Invented logarithms in 1614. Henry Briggs (1561–1630, England) — Developed the base-10 logarithm tables. Leonhard Euler (1707–1783, Switzerland) — Established the connection between logarithms, exponentials, and the number e.

A Practical Next Step

Try these three problems:

1. Simplify log⁡2(16·8) using the product rule.
2. Simplify log⁡3(81/9) using the quotient rule.
3. Compute log⁡5(125) — without a calculator.

Frequently Asked Questions

1. What are the three logarithm rules?
2. What is the change-of-base formula?
3. Is log(x+y)=log(x)+log(y)? No.
4. Why is log⁡b(1)=0? Because b⁰=1.
5. Who invented logarithms? John Napier.
6. What is the natural log ln(x)? The logarithm with base e.