Logarithm Calculator

A logarithm is the inverse operation of exponentiation. log_b(x) = y means b^y = x. In other words, it answers "to what power must b be raised to get x?". Logarithms are used in various fields including science, engineering, and statistics.

The natural logarithm has base e (Euler's number, approximately 2.71828). Written as ln(x) = log_e(x). It's frequently used in calculus and is essential for modeling natural phenomena like compound interest, population growth, and radioactive decay.

The common logarithm has base 10. Used to measure pH levels, earthquake magnitude (Richter scale), sound intensity (decibels), and more. Very useful in everyday life for converting large numbers into manageable small numbers.

Logarithms follow several important laws: the log of a product is the sum of logs (log(ab) = log(a) + log(b)), the log of a quotient is the difference of logs (log(a/b) = log(a) - log(b)), and the log of a power is the product (log(a^n) = n·log(a)).

The change of base formula allows converting a logarithm from one base to another: log_b(x) = log_a(x) / log_a(b). This enables easy calculation of logarithms with bases not provided by calculators.

Logarithms are used in many fields: entropy calculation in information theory, measuring musical scales, pH calculation in chemistry, elasticity analysis in economics, algorithm complexity analysis in computer science, and much more throughout our daily lives.