Logarithm Calculator
Calculate the logarithm of a number in base 10, base e (natural log), base 2, or any custom base.
So funktioniert's
A logarithm answers the question "what power do I need to raise this base to, to get this number?" â log_b(x) = y means b^y = x. This calculator uses the change-of-base formula, log_b(x) = ln(x) / ln(b), to support any base, with direct shortcuts for the most common bases: 10, e, and 2.
Base-10 logarithms (often written simply as "log") are useful for understanding orders of magnitude â log(100) = 2 because 10ÂČ = 100, and log(1,000,000) = 6 because 10ⶠ= 1,000,000. This makes base-10 logs a natural fit for scales like the Richter scale (earthquake magnitude) and the decibel scale (sound intensity), both of which compress huge ranges of values into manageable numbers.
Natural logarithms (written "ln") use base e (approximately 2.71828), a mathematical constant that arises naturally in calculus, continuous compound growth, and decay processes. Natural log is the standard choice in science and higher mathematics because many growth and decay formulas simplify most cleanly when expressed in base e.
Base-2 logarithms are the standard in computer science, since binary systems are built on powers of 2 â logâ(x) answers "how many times do I need to double 1 to reach x," which is directly useful for measuring things like the number of bits needed to represent a value or the number of steps in a binary search algorithm.
Logarithms are only defined for positive real numbers, since no real exponent applied to a positive base can ever produce zero or a negative result â a positive base raised to any power stays positive. This is why log(0) and log of negative numbers have no real-number answer, and this calculator requires positive input accordingly.
Logarithms are the mathematical inverse of exponentiation, the same way subtraction is the inverse of addition and division is the inverse of multiplication â this relationship is what lets logarithms "undo" exponential growth, turning a fast-growing curve into a straight or gently curving line that's much easier to read and compare on a graph.
Logarithm rules mirror exponent rules in a predictable way: the log of a product equals the sum of the logs (log(aĂb) = log(a) + log(b)), and the log of a quotient equals the difference of the logs (log(aĂ·b) = log(a) â log(b)) â these identities are why logarithms were historically used to turn difficult multiplication problems into simpler addition, long before electronic calculators existed.
Why must the input be positive?
Logarithms are only defined for positive real numbers â log(0) and log of negative numbers have no real result.
What's the difference between log and ln?
"log" conventionally means base-10 logarithm, while "ln" specifically means natural logarithm (base e, approximately 2.71828) â both are logarithms, just with different bases.
Why is base-2 logarithm important in computer science?
Binary computing is built on powers of 2, so logâ directly answers practical questions like how many bits are needed to represent a value or how many steps a binary search takes.
What does log(100) = 2 actually mean?
It means 10 raised to the power of 2 equals 100 (10ÂČ = 100) â the logarithm is simply the exponent needed to reach that value from the given base.
How does the change-of-base formula work?
log_b(x) = ln(x) Ă· ln(b) â it converts any base into a ratio of natural logarithms, which lets a calculator compute a logarithm in any base using just one underlying function.
Why do earthquake and sound intensity scales use logarithms?
Logarithmic scales compress an enormous range of raw values into a much smaller, more manageable range of numbers â each whole step represents a tenfold (or similar) increase in the underlying quantity.
Is a logarithm the opposite of an exponent?
Yes â logarithms and exponentiation are inverse operations, the same way subtraction inverts addition and division inverts multiplication.
Can I calculate a logarithm with a custom base, not just 10, e, or 2?
Yes â this calculator supports any positive base other than 1, using the change-of-base formula to compute it.
How does the log of a product relate to the logs of each factor?
log(aĂb) = log(a) + log(b) â multiplication inside a logarithm becomes addition outside it, a property historically used to simplify hand calculations before calculators existed.
Why were logarithms historically important before calculators?
Their product and quotient rules let people convert difficult multiplication and division problems into simpler addition and subtraction using logarithm tables, dramatically speeding up manual calculation.