Logarithm Calculator
Calculate natural logarithms, common logarithms, binary logarithms, and custom base logarithms with detailed step-by-step solutions and properties.
Natural Logarithm Result
Calculation Steps
Properties & Information
Common Logarithm Values
ln(e)
Natural log of e
log(100)
Common log of 100
log₂(8)
Binary log of 8
log₃(27)
Base 3 log of 27
Logarithm Properties & Rules
| Property | Formula | Example |
|---|---|---|
| Product Rule | log_b(xy) = log_b(x) + log_b(y) | log(10×100) = log(10) + log(100) = 1 + 2 = 3 |
| Quotient Rule | log_b(x/y) = log_b(x) - log_b(y) | log(100/10) = log(100) - log(10) = 2 - 1 = 1 |
| Power Rule | log_b(x^n) = n × log_b(x) | log(10²) = 2 × log(10) = 2 × 1 = 2 |
| Change of Base | log_b(x) = log_c(x) / log_c(b) | log₂(8) = ln(8) / ln(2) = 2.079 / 0.693 = 3 |
| Base Identity | log_b(b) = 1 | log₁₀(10) = 1, ln(e) = 1 |
| Zero Property | log_b(1) = 0 | log₁₀(1) = 0, ln(1) = 0 |
Common Logarithm Types
Natural Logarithm (ln)
Base e ≈ 2.71828, used in calculus and natural sciences
Common Logarithm (log)
Base 10, used in engineering and scientific notation
Binary Logarithm (log₂)
Base 2, used in computer science and information theory
Practical Applications
Finance & Economics
Compound interest, exponential growth, investment calculations
Science & Medicine
pH calculations, radioactive decay, population growth
Computer Science
Algorithm complexity, data compression, information theory
Physics & Engineering
Decibel measurements, signal processing, earthquake magnitude
Logarithm Calculator Answers
What does the logarithm calculator do?
The logarithm calculator computes logarithmic expressions for any base, including common logarithms (base 10), natural logs (base e), and custom bases.
What is a logarithm?
A logarithm answers the question: to what exponent must a base be raised to produce a given number? For example, log₁₀(100) = 2 because 10² = 100.
Can it handle natural and binary logs?
Yes. The calculator can compute natural logarithms (ln, base e), binary logs (base 2), and logs of any base you specify.
Does it show steps or properties used?
Many logarithm calculators show step-by-step solutions and apply properties like product, quotient, and power rules to simplify expressions.