Logarithm Calculator
Calculate Logarithms
Find logarithms with common bases (10, e) or custom bases. Get step-by-step solutions and explanations.
Logarithm Result
log₁₀(100) = 2
Logarithm Properties:
Logarithmic Scale:
The logarithm answers the question: "To what power must we raise the base to get the number?"
What is a Logarithm?
Logarithm is the inverse operation of exponentiation. It answers the question: "To what power must we raise the base to get the number?" For example, log₁₀(100) = 2 because 10² = 100. Logarithms are fundamental in mathematics, science, and engineering for dealing with exponential relationships.
Logarithm Types
Common Logarithm
Base 10
Most common in science
Natural Logarithm
Base e (≈2.718)
Used in calculus
Binary Logarithm
Base 2
Computer science
Custom Base
Any positive base
Flexible calculation
Logarithm Rules
1. Product Rule
The logarithm of a product is the sum of the logarithms:
logₐ(x × y) = logₐ(x) + logₐ(y)
2. Quotient Rule
The logarithm of a quotient is the difference of the logarithms:
logₐ(x / y) = logₐ(x) - logₐ(y)
3. Power Rule
The logarithm of a power is the exponent times the logarithm:
logₐ(xⁿ) = n × logₐ(x)
Real-World Applications
Science & Engineering
- Earthquake measurement: Richter scale uses logarithms
- Sound intensity: Decibel scale is logarithmic
- Chemistry: pH scale for acidity measurement
- Astronomy: Stellar magnitude brightness scale
Computer Science
- Algorithm analysis: Time complexity (O(log n))
- Data structures: Binary search trees and heaps
- Information theory: Entropy and data compression
- Cryptography: Discrete logarithm problems
Finance & Economics
- Compound interest: Calculating growth over time
- Economic modeling: Exponential growth analysis
- Stock market: Logarithmic price charts
- Risk management: Probability calculations
Biology & Medicine
- Population growth: Exponential growth models
- Drug dosage: Half-life calculations
- Epidemiology: Disease spread modeling
- Microbiology: Bacterial growth curves
Common Logarithm Examples
| Expression | Result | Explanation | Application |
|---|---|---|---|
| log₁₀(100) | 2 | 10² = 100 | Scientific notation |
| log₂(8) | 3 | 2³ = 8 | Computer science |
| ln(e) | 1 | e¹ = e | Calculus |
| log₅(125) | 3 | 5³ = 125 | General mathematics |
Important 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 |
| 100 | 2 | 4.6052 | 6.6439 |
| 1000 | 3 | 6.9078 | 9.9658 |
Step-by-Step Calculation Process
Example 1: Common Logarithm (log₁₀(1000))
- Identify base and number: base = 10, number = 1000
- Ask: "10 to what power equals 1000?"
- Recognize: 10³ = 1000
- Result: log₁₀(1000) = 3
- Verification: 10³ = 1000 ✓
Example 2: Natural Logarithm (ln(e²))
- Identify: Natural log means base e
- Apply power rule: ln(e²) = 2 × ln(e)
- Know that ln(e) = 1
- Result: 2 × 1 = 2
- Verification: e² = e² ✓
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Frequently Asked Questions (FAQs)
Q: What is the difference between log and ln?
A: log usually means log base 10 (common logarithm), while ln means log base e (natural logarithm), where e ≈ 2.71828.
Q: Can logarithms be negative?
A: Yes, logarithms can be negative when the number is between 0 and 1. For example, log₁₀(0.1) = -1 because 10⁻¹ = 0.1.
Q: What is logₐ(1)?
A: logₐ(1) = 0 for any base a, because a⁰ = 1 for all positive a.
Q: Why can't we take log of 0 or negative numbers?
A: Because there's no real number you can raise a positive base to get 0 or a negative number. Logarithms are only defined for positive real numbers.
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