Logarithm Calculator

log_b(x) = y means b^y = x. The logarithm answers: what power must I raise b to, to get x? log10(1000) = 3 because 10^3 = 1000. ln(e) = 1 because e^1 = e. log2(8) = 3 because 2^3 = 8.

The natural logarithm uses base e ≈ 2.71828. ln(x) = log_e(x). It is the inverse of e^x: ln(e^x) = x, e^(ln(x)) = x. Natural logs appear naturally in calculus, growth models, and compound interest: A = Pe^(rt).

Product: log(ab) = log(a) + log(b). Quotient: log(a/b) = log(a) - log(b). Power: log(a^n) = n*log(a). Change of base: log_b(x) = log(x)/log(b) = ln(x)/ln(b). These rules simplify multiplication into addition.

log2 is used in computer science and information theory. Bits needed to represent n values = ceil(log2(n)). Binary search takes log2(n) steps. Sorting n items takes O(n log2(n)) comparisons. log2(1,000,000) ≈ 20, so 20 bits encode 1 million values.

log (or log10) is the common logarithm, base 10. Used in pH, decibels, Richter scale. ln is the natural logarithm, base e. Used in calculus, probability, growth models. They are related: ln(x) = log(x) / log(e) ≈ log(x) / 0.4343, or ln(x) = 2.3026 x log10(x).