Log Calculator
Calculate logarithms: log base 10, natural log (ln), log base 2, and any custom base. Find antilog and inverse log values.
About the Log Calculator
A logarithm calculator computes log base 10 (common logarithm, denoted log), natural logarithm (ln, base e ≈ 2.71828), binary logarithm (log₂, base 2), and logarithms of any custom base. Logarithms are the inverse operation of exponentiation — if bˣ = y, then log_b(y) = x — and they appear throughout mathematics, science, engineering, music theory, information theory, and data analysis. The decibel scale (sound intensity), Richter scale (earthquake magnitude), pH scale (acidity), Benford's law (digit distribution), Shannon entropy (information theory), time-complexity of binary search algorithms, and musical octaves all use logarithms. Our calculator also computes antilogarithms (inverse log, finding x when log_b(x) is known), applies logarithm laws to simplify expressions, and handles the change of base formula for converting between any two bases.
Formula
log_b(x) = log(x)/log(b) | ln(x) = log_e(x) | log(ab) = log(a)+log(b) | log(aⁿ) = n×log(a)
How It Works
Definition: log_b(x) = y means b^y = x. Equivalently, log_b(x) answers "what power do I need to raise b to in order to get x?" Common log: log₁₀(1000) = 3 because 10³ = 1000. Natural log: ln(e²) = 2 because e² ≈ 7.389. Binary log: log₂(32) = 5 because 2⁵ = 32. Change of base formula: log_b(x) = ln(x)/ln(b) = log(x)/log(b). Example: log₅(125) = ln(125)/ln(5) = 4.828/1.609 = 3.0 (since 5³ = 125). Logarithm laws: log(a×b) = log(a)+log(b); log(a/b) = log(a)−log(b); log(aⁿ) = n×log(a); log_b(b) = 1; log_b(1) = 0.
Tips & Best Practices
- ✓Decibel scale: dB = 10×log₁₀(P₂/P₁) for power ratios. Every +10 dB is a 10× increase in power (but only 3.16× in amplitude). Normal conversation at 60 dB is 10⁶ times more intense than the threshold of hearing (0 dB).
Who Uses This Calculator
Physics and chemistry students calculating pH, sound intensity, and reaction kinetics. Computer scientists analysing algorithm time complexity. Mathematics students learning exponential and logarithmic functions. Engineers working with decibel calculations in audio and communications. Finance students applying logarithms in continuous compounding and option pricing. Biologists using logarithmic transformations for skewed biological data. Students solving exponential equations by applying logarithms.
Optimised for: USA · Canada · UK · Australia · Calculations run in your browser · No data stored
Frequently Asked Questions
What is log base 10?
Log base 10 (common log) asks: 10 to what power equals x? log₁₀(1000) = 3 because 10³ = 1000.