Logarithm Calculator
Calculate logarithms with any base, including common log (base 10), natural log (ln, base e), and antilogarithms. See results in multiple bases simultaneously.
Logarithm Calculator
log₁₀(x) = ?
How to Use the Logarithm Calculator
Select the type of logarithm you want to calculate, enter the value, and click Calculate. For custom bases, enter the base number first.
Types of Logarithms
Common Logarithm (log₁₀)
The common logarithm uses base 10 and is often written as "log" without a subscript. It answers: "10 raised to what power equals x?"
Natural Logarithm (ln)
The natural logarithm uses base e (≈ 2.71828) and is written as "ln". It's fundamental in calculus, physics, and many scientific applications.
Logarithm with Any Base
Calculate log_b(x) for any positive base b (where b ≠ 1). Uses the change of base formula: log_b(x) = ln(x) / ln(b).
Antilogarithm
The antilogarithm is the inverse operation. If log_b(x) = y, then antilog_b(y) = b^y = x.
Logarithm Laws
- 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_b(x) = log_c(x) / log_c(b)
Real-World Applications
- Sound intensity measurement (decibels)
- Earthquake magnitude (Richter scale)
- Acidity measurement (pH scale)
- Compound interest and exponential growth
- Information theory and data compression
- Algorithm complexity analysis (computer science)