Logarithm Calculator
Calculate the logarithm of a number in any base, plus the natural log (ln) and common log (log₁₀), and go the other way with exponents.
Comment utiliser cette calculatrice
- 1Choose whether you want a logarithm or its inverse (an exponent).
- 2Enter the number, and the base — 10 for common log, e for natural log, 2 for binary.
- 3Read the result along with the natural, common, and binary logs.
Comment ça marche
Logarithms
log_b(x) = y means bʸ = x change of base: log_b(x) = ln(x) ÷ ln(b) log(a × c) = log(a) + log(c) log(aⁿ) = n × log(a) natural log uses base e ≈ 2.71828; common log uses base 10
A logarithm is the inverse of exponentiation: log_b(x) is the power to which the base b must be raised to produce x. Its defining property — that the log of a product is the sum of the logs — turns multiplication into addition, which is why logarithms were the foundation of hand computation for three centuries and why slide rules worked. Any base can be reached from any other via the change-of-base formula, so a calculator only needs one natural log to compute them all.
Exemple détaillé
log₁₀(1000) = 3, because 10³ = 1000 — the common log of a power of ten is just the number of zeros. In base 2, log₂(1000) ≈ 9.97, meaning 1000 is just under 2¹⁰ (1024). The natural log, ln(1000) ≈ 6.91.
Logarithm Calculator : le guide complet
What a logarithm actually asks
A logarithm answers a single question: to what power must I raise the base to get this number? log₁₀(1000) = 3 because 10 raised to the 3rd power is 1000. That is all a log is — the exponent, extracted. Once you read it that way, the notation stops being intimidating: the base is what you're multiplying, the answer is how many times.
This makes logarithms and exponents two sides of one relationship. If bˣ = y, then log_b(y) = x, and vice versa. The 'antilog' mode above is simply exponentiation, included because undoing a log is exactly as common as taking one. Recognising this inverse pairing is the key to using logarithms confidently rather than memorising rules.
The three bases you'll actually meet
Although a logarithm can use any base, three dominate. Base 10 (the common log) matches our decimal system — the log of a power of ten is just its number of zeros — and it underlies decibels, pH, and the Richter scale. Base e ≈ 2.718 (the natural log, written ln) arises whenever something grows or decays continuously, from compound interest to radioactive decay, because e is the natural constant of continuous growth. Base 2 (the binary log) counts doublings and is everywhere in computing.
The change-of-base formula ties them together: any log equals the natural log of the number divided by the natural log of the base. This is why a calculator with only ln and log₁₀ buttons can still compute a log in base 7 — and it is exactly what this tool does internally, reporting all three common bases at once so you never have to convert by hand.
Why logarithms tame huge ranges
Logarithms compress enormous spans of numbers into manageable ones, which is why so many scientific scales are logarithmic. Earthquake energy spans a factor of billions, so the Richter scale takes its log — a magnitude 6 releases about 32 times the energy of a 5, not 20% more. Sound intensity, acidity, and stellar brightness use the same trick, turning multiplicative jumps into evenly spaced steps.
The historical reason logarithms were invented is the property that log(a × b) = log(a) + log(b). Before calculators, this let people multiply large numbers by looking up their logs, adding, and reversing — the entire purpose of logarithm tables and the slide rule. That addition-instead-of-multiplication shortcut also explains why logarithmic graphs turn exponential growth into a straight line, making otherwise unreadable data easy to interpret.
Questions fréquentes
What is a logarithm?
The power to which a base must be raised to produce a number. log₁₀(1000) = 3 because 10³ = 1000. It is the inverse of exponentiation: if bˣ = y, then log_b(y) = x.
What is the difference between ln and log?
ln is the natural logarithm, base e (≈ 2.718), which appears in continuous growth and decay. 'log' usually means the common logarithm, base 10, used in decibels, pH, and the Richter scale. This calculator computes both, plus any base you enter.
How do I calculate a log in a base my calculator doesn't have?
Use the change-of-base formula: log_b(x) = ln(x) ÷ ln(b), or equally with log₁₀. Divide the natural log of the number by the natural log of the base. This calculator applies it automatically for any base you enter.
Why can't I take the log of zero or a negative number?
Because no power of a positive base ever produces zero or a negative number — bˣ is always positive. So the logarithm is only defined for positive inputs. As x approaches zero the log heads to negative infinity, but it never reaches zero itself.