The Log Calculator finds the logarithm of a number in any base, including e (natural log), and can also solve for the number or the base.
log(100) = 4.605170185988
| log base e of 100 | 4.605170185988 |
| Natural log ln | 4.605170185988 |
| Common log log₁₀ | 2 |
| Binary log log₂ | 6.643856189775 |
A logarithm answers "to what power must the base be raised to get this number?" log_b(x) = y means bʸ = x. For example log₁₀(1000) = 3 because 10³ = 1000. Logarithms turn multiplication into addition, which made them essential for calculation before computers.
Base 10 (common log, log) is used in science and engineering scales. Base e ≈ 2.71828 (natural log, ln) appears in calculus, growth and decay. Base 2 (binary log) is used in computer science and information theory.
Worked example (defaults): ln(100) = 4.605170186, because e^4.60517 ≈ 100.
log(xy) = log x + log y. log(x/y) = log x − log y. log(xⁿ) = n log x. Change of base: log_b(x) = ln x ÷ ln b. log_b(1) = 0 and log_b(b) = 1.
The Richter and moment magnitude scales for earthquakes, decibels for sound, pH for acidity, star brightness, compound interest time calculations and algorithm complexity (O(log n)) all use logarithms.
Can you take the log of a negative number? Not in real numbers.
What is log of 0? Undefined (it tends to minus infinity).
Why can't the base be 1? Because 1 raised to any power is 1.