Logarithm Calculator
Calculate logarithms with any base including natural log (ln), common log (log₁₀), and binary log (log₂) with detailed analysis.
Logarithm Calculator
log₍10₎(100)
About Logarithms
A logarithm is the inverse operation of exponentiation. If b^x = y, then log_b(y) = x. Logarithms are fundamental in mathematics, science, and engineering.
Types of Logarithms
- Common Logarithm (log₁₀): Base 10, used in scientific calculations
- Natural Logarithm (ln): Base e ≈ 2.71828, used in calculus
- Binary Logarithm (log₂): Base 2, used in computer science
- Custom Base: Any positive base except 1
Logarithm Properties
- Product Rule: log_b(xy) = log_b(x) + log_b(y)
- Quotient Rule: log_b(x/y) = log_b(x) - log_b(y)
- Power Rule: log_b(x^n) = n × log_b(x)
- Change of Base: log_b(x) = log_c(x) / log_c(b)
- Inverse Property: b^(log_b(x)) = x
Special Values
- log_b(1) = 0 (for any base b)
- log_b(b) = 1 (logarithm of base equals 1)
- log_b(b^n) = n (logarithm of base raised to power)
- ln(e) = 1 (natural log of e)
Applications
- Science: pH calculations, decibel measurements, Richter scale
- Finance: Compound interest, time value of money
- Computer Science: Algorithm complexity, information theory
- Engineering: Signal processing, control systems
- Statistics: Log-normal distributions, regression analysis
Common Logarithm Values
- log₁₀(10) = 1
- log₁₀(100) = 2
- log₁₀(1000) = 3
- ln(e) = 1
- ln(e²) = 2