Log Calculator (Logarithm)
Calculate logarithms, antilogs, and bases with step-by-step solutions
Logarithm Calculator
Result
Step-by-Step Solution
Logarithm Types
Natural (ln)
Base e ≈ 2.718281
Used in calculus, growth models
Common (log)
Base 10
Used in science, engineering
Binary (log₂)
Base 2
Used in computer science
Custom
Any positive base ≠ 1
Uses change of base formula
Key Properties
Product Rule
log(xy) = log(x) + log(y)
Quotient Rule
log(x/y) = log(x) - log(y)
Power Rule
log(x^n) = n × log(x)
Change of Base
log_a(x) = log(x) / log(a)
Special Values
log(1) = 0, log(base) = 1
Understanding Logarithms
What is a Logarithm?
A logarithm is the inverse operation of exponentiation. If b^y = x, then log_b(x) = y. In other words, the logarithm tells us what power we need to raise the base to get a specific number.
Key Relationships
- •If b^y = x, then log_b(x) = y
- •b^(log_b(x)) = x (for x > 0)
- •log_b(b^x) = x
- •Logarithms are only defined for positive numbers
Common Logarithm Values
| x | log₁₀(x) | ln(x) |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 0.301 | 0.693 |
| e ≈ 2.718 | 0.434 | 1 |
| 10 | 1 | 2.303 |
| 100 | 2 | 4.605 |
| 1000 | 3 | 6.908 |
Change of Base Formula
To calculate logarithms with any base using natural or common logarithms:
Real-World Applications
Science & Engineering
pH scale (acidity), Richter scale (earthquakes), decibel scale (sound)
Finance
Compound interest calculations, investment growth, inflation models
Computer Science
Algorithm complexity, binary logarithms, information theory