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Grade 12 Advanced Functions, University Preparation (MHF4U)
A. Exponential and Logarithmic Functions
2. 12AF.A1.2
2.2 Natural log: ln
What Is Natural Log (ln)?
Natural log, written ln, is the logarithm base e. See its definition, formula, graph, and how it connects to exponential functions.
What You'll Learn
The natural log, written ln(x), is the logarithm with base e, where e is approximately 2.71828
ln(x) answers the question: e raised to what power gives x
ln(1) always equals 0, and ln(e) always equals 1
The domain of ln(x) is all positive real numbers, and its graph passes through (1, 0) with a vertical asymptote at x equals 0
ln is sometimes called the Naperian logarithm, and it differs from the common logarithm, which uses base 10
What You'll Practice
1
Evaluating natural log expressions like ln(e) without a calculator
2
Simplifying expressions with e raised to natural log powers
3
Using the inverse relationship between e and ln to solve problems
Why This Matters
Natural logarithms are essential in calculus, physics, and engineering. You'll use ln when studying exponential growth and decay, compound interest, and advanced calculus. Mastering the relationship between e and ln makes complex higher-level math much more manageable.
Before You Start — Make Sure You Can:
This Unit Includes
3 Video lessons
Practice exercises
Learning resources
Skills
Natural Logarithm
Logarithms
Base e
Exponential Form
Inverse Properties
Logarithmic Rules
ON Curriculum Aligned