What is a logarithm?
The logarithm answers: "what exponent turns the base into this number?" log₁₀(1000) = 3 because 10³ = 1000. Logarithms are the inverse of exponentiation, turning multiplication into addition — the property that made slide rules work and still powers decibels, pH, and earthquake magnitude scales.
What is the difference between ln, log₁₀, and log₂?
They differ only in base: ln uses e ≈ 2.71828 (natural sciences, calculus, continuous growth), log₁₀ uses 10 (engineering, pH, decibels), and log₂ uses 2 (computer science, information theory — "how many times can you halve this?"). Any log can be converted to another base by dividing: log_b(x) = ln(x)/ln(b).
Why can't I take the log of a negative number or zero?
Because no real exponent turns a positive base into a negative number or zero — 10 to any power is always positive. log(0) approaches negative infinity, and log of negatives requires complex numbers (beyond this calculator's scope). The domain of every real logarithm is strictly positive numbers.
What is the change-of-base formula used for?
log_b(x) = ln(x) / ln(b) lets you compute a logarithm in any base using only the ln or log₁₀ button on a basic calculator. Need log₇(50)? Compute ln(50)/ln(7) ≈ 2.011. This calculator applies it automatically for whatever base you enter.
Where do logarithms turn up outside a maths class?
Almost anywhere a quantity spans an enormous range and needs compressing into a usable scale. pH is the negative log of hydrogen ion concentration, so pH 4 is ten times more acidic than pH 5. Decibels are logarithmic, which is why 90 dB is not slightly worse than 80 dB but ten times the sound intensity. Earthquake magnitude, stellar brightness, camera stops, and the semitone spacing of musical pitch all work the same way. In computing, information is measured in bits — a base-2 logarithm of the number of possible states — and an algorithm described as O(log n) is one that barely slows down as the data grows.
What are the logarithm rules and why do they work?
Three identities do most of the work: log(ab) equals log a plus log b, log(a divided by b) equals log a minus log b, and log(a to the power n) equals n times log a. They follow directly from the exponent rules, because a logarithm is just an exponent written the other way round. Their practical effect is to turn multiplication into addition and powers into multiplication, which is precisely what made log tables and slide rules so valuable for three centuries before electronic calculators. The same trick is still used today to keep very small probabilities from underflowing to zero in statistical software.