Log Calculator

A logarithm is the inverse operation of exponentiation. log₂(x) = y means b^y = x. Example: log₁₀(100) = 2 because 10² = 100. Logarithms answer: "What power do I need to raise this base to get this number?" They convert multiplication into addition: log(a×b) = log(a) + log(b), which made them essential before calculators for astronomical and navigational calculations.

"log" usually means log base 10 (common logarithm). "ln" means log base e (natural logarithm), where e ≈ 2.71828. log₁₀(10) = 1. ln(e) = 1. Both satisfy log(1) = 0. In pure mathematics and calculus, "log" often means ln. In engineering and practical science, "log" means log₁₀. Always check context. They are related by: log₁₀(x) = ln(x) / ln(10) = ln(x) / 2.302585.

e = 2.71828182845... is an irrational constant that arises naturally in mathematics. It is the base of the natural logarithm. Key property: d/dx(e^x) = e^x — e^x is its own derivative, making it the natural base for exponential growth and decay. It appears in: compound interest (continuously compounded), population growth, radioactive decay, the bell curve (normal distribution), and Euler's identity e^(iπ) + 1 = 0, called the most beautiful equation in mathematics.

Product rule: log(a×b) = log(a) + log(b). Quotient rule: log(a/b) = log(a) − log(b). Power rule: log(a^n) = n × log(a). Change of base: log₂(x) = log(x)/log(b). Special values: log₂(1) = 0, log₂(b) = 1. These rules allow complex logarithm calculations to be broken into simpler parts. Example: log(1000×100) = log(1000) + log(100) = 3 + 2 = 5.

Change of base: log₂(x) = log_a(x) / log_a(b). Most commonly: log₂(x) = ln(x)/ln(b) = log(x)/log(b). Example: log₅(25) = log(25)/log(5) = 1.39794/0.69897 = 2. Verify: 5² = 25. This is how calculators with only log and ln buttons can compute logarithms of any base. It works because both sides of the formula equal the same exponent.

The antilog (antilogarithm) is the inverse of a logarithm. If log₁₀(x) = y, then antilog₁₀(y) = 10^y = x. Example: log₁₀(x) = 2.5. x = 10^2.5 ≈ 316.23. For natural log: if ln(x) = y, then antiln(y) = e^y = x. Antilogs are used to convert log-scale results back to linear scale, for example when reading pH scales, decibel levels, or Richter scale values.

pH scale (acidity): pH = −log₁₀[H+]. Each pH unit = 10x change in acidity. Decibels (sound): dB = 10 × log₁₀(power ratio). Richter scale (earthquakes): each unit = 10x more ground motion. Musical frequency: doubling frequency = 1 octave (log₂ scale). Algorithm complexity: O(log n) for binary search. Computer storage: 2^10 = 1,024 ≈ 1K. Wherever values span many orders of magnitude, a log scale makes them manageable.

Log base 2 (log₂ or lg) is fundamental in computer science and information theory. It answers: "How many times must I divide by 2 to reach 1?" log₂(1024) = 10, meaning 1024 = 2^10 = 1 kilobyte. Uses: binary search complexity O(log₂ n), information entropy (bits of information = log₂ of possible outcomes), musical intervals, and comparing exponents in algorithm analysis. Claude Shannon founded information theory using log₂.

Not in real numbers. log(0) is undefined (approaches −∞ as x approaches 0 from the right). log(negative number) is undefined in real numbers because no real power of a positive base gives a negative result. In complex number mathematics, logarithms of negative numbers are defined but yield complex results: ln(−1) = iπ (Euler's formula). This calculator works with positive real numbers only.

log₂(1) = 0 for any valid base b. This is because b^0 = 1 for any non-zero b. So log₁₀(1) = 0, ln(1) = 0, log₂(1) = 0, log₅(1) = 0. Also: log₂(b) = 1 for any base b (since b^1 = b). These two facts — log(1) = 0 and log(base) = 1 — are the anchor points for understanding any logarithm.