Calculate logarithm of any number in any base. Free log calculator.
This logarithm tool lets you compute natural logarithm (ln), common logarithm (log base 10), and logarithm of any custom base. It is designed for speed and accuracy, running entirely in your browser with no data sent to any server. Whether you are a student, professional, or simply need a quick answer, this calculator provides instant results with clear explanations. The calculator uses the change of base formula: log_b(x) = ln(x) / ln(b). For natural log, it computes ln directly. Results are displayed with up to six significant digits.
The calculator uses the change of base formula: log_b(x) = ln(x) / ln(b). For natural log, it computes ln directly. Results are displayed with up to six significant digits.
Students solve logarithmic equations for math and science courses. Engineers use logarithms in signal processing and decibel calculations. Scientists analyze exponential growth and decay data. Computer science students understand algorithm complexity. Finance professionals calculate investment doubling times.
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A logarithm answers the question: to what power must a base be raised to produce a given number? For example, log base 2 of 8 equals 3, because 2 raised to the power of 3 equals 8. Logarithms are the inverse operation of exponentiation.
Log typically refers to the common logarithm with base 10 (log10). Ln refers to the natural logarithm with base e (approximately 2.71828). In mathematics and science, ln is more commonly used because the constant e appears naturally in calculus and exponential growth formulas.
The change of base formula lets you calculate a logarithm with any base using natural or common logarithms: log_b(x) = ln(x) / ln(b) = log(x) / log(b). This is how calculators compute logarithms of arbitrary bases.
In real number mathematics, logarithms of negative numbers are undefined because no real power of a positive base produces a negative result. In complex number mathematics, logarithms of negative numbers do exist but produce complex values.
Logarithmic scales measure earthquake intensity (Richter scale), sound loudness (decibels), acidity (pH), and stellar brightness. In finance, logarithms calculate compound interest periods. In computer science, logarithms describe algorithm efficiency and binary search performance.