Log Calculator
Calculate logarithms, natural logs, and antilogs with step-by-step solutions
Logarithm Definition
If by = x, then logb(x) = y
Calculator
Calculate log with a custom base.
Calculate natural log, common log, and binary log simultaneously.
x =
Find the antilogarithm (inverse of log): by = x
y =
Explore logarithm properties with your values.
Results
Understanding Logarithms
A logarithm answers the question: “How many of one number do we multiply to get another number?” For example, log₁₀(100) = 2 because 10 × 10 = 100, or 10² = 100.
Common Logarithm Types
- Natural Log (ln): Base e (≈ 2.71828) — used in calculus, physics, and continuous growth
- Common Log (log): Base 10 — used in science, engineering, and pH calculations
- Binary Log (lb): Base 2 — used in computer science and information theory
Key Logarithm Properties
Product Rule
logₐ(M × N) = logₐ(M) + logₐ(N)
Quotient Rule
logₐ(M / N) = logₐ(M) − logₐ(N)
Power Rule
logₐ(Mp) = p × logₐ(M)
Change of Base
logₐ(b) = logₙ(b) / logₙ(a)
Real-World Applications
- Science: pH scale, Richter scale for earthquakes, decibel scale for sound
- Finance: Compound interest calculations and investment growth models
- Computer Science: Algorithm complexity (Big O notation), data compression
- Engineering: Signal processing, filter design, control systems
- Medicine: Drug dosage calculations and bacterial growth rates