Calculate logarithms in any base with antilog, change of base formula, and key properties.
Quick Answer:Enter a number and select a base to instantly calculate its logarithm. Results shown for log base 2, e (ln), 10, and your custom base simultaneously.
Logarithm Result
Calculating...
log base 2
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ln (base e)
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log base 10
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Antilog (base^result)
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Logarithmic scales are everywhere in daily life: the Richter scale for earthquakes (each point is 10x more energy), decibels for sound (logarithmic intensity), and pH for acidity. Understanding that log properties convert multiplication into addition is the key insight -- this is why slide rules worked and why logarithms revolutionized computation before electronic calculators.
log (common logarithm) uses base 10 and is widely used in engineering and science. ln (natural logarithm) uses base e (approximately 2.71828) and is fundamental in calculus and continuous growth models. log base 2 (binary logarithm) is essential in computer science for analyzing algorithms and data structures. They are all related through the change of base formula: log_b(x) = ln(x) / ln(b).
The three key properties are: (1) Product rule: log(a*b) = log(a) + log(b), (2) Quotient rule: log(a/b) = log(a) - log(b), (3) Power rule: log(a^n) = n*log(a). Additionally, log_b(b) = 1, log_b(1) = 0, and the change of base formula log_b(x) = log_c(x) / log_c(b) lets you convert between any bases.
Logarithms are undefined for negative numbers and zero in the real number system because no real power of a positive base can produce a negative number or zero. For example, there is no real x such that 10^x = -5. In the complex number system, logarithms of negative numbers do exist using Euler's formula, resulting in complex values involving pi*i.