Calculate Logarithms — Any Base
Calculate log base 10, natural log (ln), or any custom base. Shows step-by-step conversion between bases.
Understanding Logarithms
A logarithm answers: "what power do I raise the base to, to get this number?" log₁₀(100) = 2 because 10² = 100. ln(e) = 1 because e¹ = e. log₂(256) = 8 because 2⁸ = 256. Logarithms are the inverse of exponents — they "undo" exponential growth. This makes them essential for measuring earthquakes (Richter scale), sound (decibels), and pH (acidity).
Common Logarithm Bases
Base 10 (log): Used in science, engineering, pH scale. log(1000) = 3. Base e (ln): Natural logarithm, used in calculus, compound interest, growth/decay. ln(1) = 0, ln(e) = 1. Base 2: Used in computer science (bits, binary). log₂(1024) = 10. Change of base formula: log_b(x) = ln(x)/ln(b) — converts between any bases.
Logarithmic scales are everywhere in daily life: the Richter scale (each whole number = 10x more energy), decibels (every 10 dB = 10x louder), and pH (each unit = 10x more acidic). Understanding logs means understanding these scales intuitively.
