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Algebra, Exponents & Roots • Lesson 21 ⏱ 6 min read 📅 Updated Nov 2026

Factorials (n!) and Logarithms (log & ln): Concepts and Practical Rules

Understand what factorial n! means, why 0! = 1, how logarithms undo exponentiation, and the key properties of common log and natural ln.

Core Concept

A factorial (( n! )) counts permutations (multiplying decreasing integers). A logarithm is the inverse of exponentiation, asking: “To what power must base b be raised to get x?”

4 × 3 × 2 × 1 4! = 4 × 3 × 2 × 1 = 24
Factorial Multiplying Chain: Count Ways to Sequence Items

Factorials Reference (0! to 6!)

FactorialExpansionValueWhere It Is Used
0!Defined foundation1Combinatorics formulas
1!11Single item arrangement
3!3 × 2 × 16Arranging 3 books on a shelf
4!4 × 3 × 2 × 1244-digit permutations
5!5 × 24120Deck shuffling probabilities
6!6 × 120720Password security math
Try It Yourself

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Solved Step-by-Step Examples:Factorial of 6 (6!)Logarithm Calculator
Your Turn!

Check Your Understanding (3 Quick Questions)

Score: 0 / 3

Question 1: What is 5! (5 factorial)?

Question 2: By definition in mathematics, what is 0!?

Question 3: What is the value of log₁₀(1,000)?