Logarithm Calculator
Calculate the logarithm of any number with any base. Supports common logarithms (base 10), natural logarithms (base e), binary logarithms (base 2), and custom bases.
Calculate Logarithm
Enter a positive number and base to calculate the logarithm:
Change of base formula: logb(x) = ln(x) ÷ ln(b)
What is a Logarithm?
A logarithm is the inverse operation of exponentiation. If b^y = x, then log_b(x) = y. The logarithm tells you what exponent you need to raise the base to in order to get the number.
Mathematical Notation
means by = x
Common Logarithms
📊 Common Logarithm
Base 10: log₁₀(x)
Example: log₁₀(100) = 2 because 10² = 100
🌿 Natural Logarithm
Base e: ln(x)
Example: ln(e) = 1 because e¹ = e
💻 Binary Logarithm
Base 2: log₂(x)
Example: log₂(8) = 3 because 2³ = 8
Logarithm Rules (Laws)
📈 Product Rule
log_b(x × y) = log_b(x) + log_b(y)
Example: log₁₀(100 × 1000) = log₁₀(100) + log₁₀(1000) = 2 + 3 = 5
➗ Quotient Rule
log_b(x ÷ y) = log_b(x) - log_b(y)
Example: log₁₀(1000 ÷ 10) = log₁₀(1000) - log₁₀(10) = 3 - 1 = 2
⚡ Power Rule
log_b(x^n) = n × log_b(x)
Example: log₂(8³) = 3 × log₂(8) = 3 × 3 = 9
🔄 Change of Base
log_b(x) = log_c(x) ÷ log_c(b)
Example: log₂(8) = ln(8) ÷ ln(2) ≈ 2.079 ÷ 0.693 ≈ 3
Special Values
| Expression | Value | Reason |
|---|---|---|
| log_b(1) | 0 | b⁰ = 1 for any base b |
| log_b(b) | 1 | b¹ = b |
| log_b(b^n) | n | b^n = b^n |
| log₁₀(10) | 1 | 10¹ = 10 |
| ln(e) | 1 | e¹ = e |
| log₂(2) | 1 | 2¹ = 2 |
Applications of Logarithms
🧪 pH Scale
pH = -log₁₀[H⁺]
Measures acidity of solutions
🌋 Richter Scale
Earthquake magnitude uses logarithmic scale
🎵 Decibel Scale
dB = 10 × log₁₀(P₁/P₀)
Sound intensity measurement
💻 Information Theory
Entropy calculations use log₂
Exponential vs Logarithmic Relationship
| Exponential Form | Logarithmic Form | Example |
|---|---|---|
| 2³ = 8 | log₂(8) = 3 | 2 to what power equals 8? |
| 10² = 100 | log₁₀(100) = 2 | 10 to what power equals 100? |
| e^0 = 1 | ln(1) = 0 | e to what power equals 1? |
| 5^2 = 25 | log₅(25) = 2 | 5 to what power equals 25? |
💡 Pro Tip: Logarithms are extremely useful for working with very large or very small numbers. They convert multiplication and division into addition and subtraction, making complex calculations much simpler!
Logarithm Calculation Methods
- Direct Calculation: Using built-in functions for common bases (log₁₀, ln, log₂)
- Change of Base: Convert any logarithm to natural logarithms using the formula log_b(x) = ln(x)/ln(b)
- Scientific Calculator: Most calculators have dedicated logarithm buttons
- Log Tables: Historical method using pre-computed logarithm tables
Common Logarithm Values
| Number | log₁₀(x) | ln(x) | log₂(x) |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.3010 | 0.6931 | 1 |
| 10 | 1 | 2.3026 | 3.3219 |
| e (≈2.718) | 0.4343 | 1 | 1.4427 |
| 100 | 2 | 4.6052 | 6.6439 |
| 1000 | 3 | 6.9078 | 9.9658 |