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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:

| Rule | Identity | Effect |
| --- | --- | --- |
| **Product rule** | log⁡b(xy)=log⁡b(x)+log⁡b(y) | Multiplication → addition |
| **Quotient rule** | log⁡b!(xy)=log⁡b(x)−log⁡b(y) | Division → subtraction |
| **Power rule** | log⁡b(xn)=nlog⁡b(x) | Exponentiation → multiplication |

## 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.