Calculate Natural Log

Enter a positive number to calculate its natural logarithm:

Natural Log Formula:
ln(x) = logarithm with base e
e ≈ 2.71828 (Euler's number)

Number (x)

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Show Mathematical Properties

Understanding Natural Logarithms

The natural logarithm, denoted as ln(x), is one of the most important functions in mathematics and science. It uses Euler's number e (approximately 2.71828) as its base and appears in countless natural phenomena and mathematical models.

What is the Natural Logarithm?

📐 Definition

ln(x) is the power to which e must be raised to obtain x
If ln(y) = x, then e^x = y
This makes ln and e^ inverse functions

🎯 Euler's Number (e)

e ≈ 2.718281828459045...
An irrational number discovered by Leonhard Euler
The base of the natural logarithm

🔍 Key Properties

ln(1) = 0 (any number to power 0 = 1)
ln(e) = 1 (e to power 1 = e)
ln(e^x) = x (inverse function property)

💡 Pro Tip: Natural logarithms are called "natural" because they arise naturally in many areas of mathematics, physics, and biology. Unlike logarithms with other bases (like base 10), natural logs appear in the solutions to many differential equations.

Common Natural Log Values

Number (x)	ln(x)	Description	Verification (e^ln(x))
1	0	ln(1) = 0	e⁰ = 1 ✓
2	0.693	ln(2) ≈ 0.693	e^0.693 ≈ 2 ✓
e (≈2.718)	1	ln(e) = 1	e¹ = e ✓
10	2.302	ln(10) ≈ 2.302	e^2.302 ≈ 10 ✓
0.5	-0.693	ln(0.5) ≈ -0.693	e^(-0.693) ≈ 0.5 ✓
100	4.605	ln(100) = 2×ln(10) ≈ 4.605	e^4.605 ≈ 100 ✓

Logarithm Properties

➕ Multiplication Rule

ln(a × b) = ln(a) + ln(b)
Example: ln(6) = ln(2×3) = ln(2) + ln(3)
≈ 0.693 + 1.099 = 1.792

➖ Division Rule

ln(a ÷ b) = ln(a) - ln(b)
Example: ln(4) = ln(8÷2) = ln(8) - ln(2)
≈ 2.079 - 0.693 = 1.386

⚡ Power Rule

ln(a^b) = b × ln(a)
Example: ln(8) = ln(2³) = 3 × ln(2)
≈ 3 × 0.693 = 2.079

Real-World Applications

🔬 Physics & Chemistry

Radioactive decay: N = N₀e^(-λt)
pH calculations: pH = -ln[H⁺]
Thermodynamics: ΔG = -RT ln K
Reaction rates: k = Ae^(-Ea/RT)

💰 Finance & Economics

Continuous compounding: A = Pe^(rt)
Economic growth models
Investment returns
Inflation calculations

⚕️ Biology & Medicine

Population growth: P = P₀e^(rt)
Drug concentration modeling
Bacterial growth studies
Epidemiology models

🔧 Engineering

Heat transfer equations
Electrical circuit analysis
Vibration and resonance
Control system design

Exponential vs Logarithmic Relationships

Exponential Form	Logarithmic Form	Example	Application
e^x = y	ln(y) = x	e^2 ≈ 7.389 ln(7.389) ≈ 2	Growth models
e^(-kt) = y	ln(y) = -kt	e^(-0.1×5) ≈ 0.607 ln(0.607) ≈ -0.5	Decay processes
e^(rt) = y	ln(y) = rt	e^(0.05×10) ≈ 1.649 ln(1.649) ≈ 0.5	Compound interest
e^(-Ea/RT) = k	ln(k) = -Ea/RT	e^(-50000/2500) ≈ 0.135 ln(0.135) ≈ -2.0	Chemical kinetics

Comparison with Other Logarithms

📏 Common Log (log₁₀)

Base 10 logarithm
log₁₀(x) = log(x)
Used in scientific notation
Example: log₁₀(100) = 2

🔢 Binary Log (log₂)

Base 2 logarithm
Used in computer science
Example: log₂(8) = 3
Important for algorithms

📈 Natural Log (ln)

Base e logarithm
Most fundamental in calculus
Appears in differential equations
Example: ln(e) = 1

Conversion Between Logarithm Bases

Conversion Formula	Example	Calculation
log₁₀(x) = ln(x) / ln(10)	log₁₀(1000) = ?	ln(1000)/ln(10) ≈ 6.908/2.302 ≈ 3
log₂(x) = ln(x) / ln(2)	log₂(16) = ?	ln(16)/ln(2) ≈ 2.773/0.693 ≈ 4
ln(x) = log₁₀(x) × ln(10)	ln(100) = ?	2 × 2.302585 ≈ 4.605

Advanced Applications

🧮 Calculus & Mathematics

Derivative: d/dx[ln(x)] = 1/x
Integral: ∫ ln(x) dx = x ln(x) - x + C
Appears in solutions to differential equations

🔬 Scientific Computing

Numerical analysis algorithms
Statistical modeling
Data analysis and fitting
Optimization problems

📊 Probability & Statistics

Log-normal distributions
Maximum likelihood estimation
Information theory
Entropy calculations

Natural Log Calculator