Logarithm Calculator

A logarithm answers one question: to what power must a given base be raised to produce a particular number? Log base 10 of 1000 is 3, because 10^3 = 1000, and in general log_b(x) = y says the same thing as b^y = x. That makes log₁₀(100) equal to 2, since 10² = 100, and log₂(8) equal to 3, since 2³ = 8. Logarithms are exponents read backwards, which is why the two topics are always taught alongside each other.

Three bases account for nearly all practical use. Base 10 gives the common logarithm, and on most UK calculators the button marked 'log' means log₁₀. Base e, where e is roughly 2.718, gives the natural logarithm, written ln. Base 2 turns up throughout computing. This calculator handles all three plus any custom base you enter, converting between them with the change-of-base formula log_b(x) = ln(x) / ln(b). A-Level Maths uses common and natural logs side by side, so moving between the two is a routine skill rather than an advanced one.

The log laws exist because they turn hard operations into easy ones. Multiplication becomes addition, since log(xy) = log(x) + log(y). Division becomes subtraction, since log(x/y) = log(x) − log(y). Powers come down to the front, so log(x^n) = n × log(x). Two identities complete the set: log(1) = 0 in any base, and log_b(b) = 1. The change-of-base rule then lets you compute a logarithm in any base at all from the base 10 or base e buttons you actually have in front of you.

Logarithmic scales appear wherever quantities span enormous ranges. The Richter scale is one, which is why a magnitude 7 earthquake is 10 times larger than a magnitude 6. Decibels are another, where a 10 dB increase means 10 times louder. The pH scale behaves the same way, making pH 5 ten times more acidic than pH 6. Star brightness uses a magnitude scale of the same type, information theory measures choices in bits, where the count is log₂ of the number of possibilities, and computer scientists describe algorithmic complexity in the same terms.

Natural logarithms deserve separate attention, because e, at about 2.71828, arises on its own in anything growing continuously. Its defining property is that the derivative of e^x is e^x, while the derivative of ln x is 1/x, which is why calculus is full of both. Continuous compound interest follows A = Pe^(rt), and the same exponential form describes radioactive decay, population growth and Newton's law of cooling. A-Level Maths gives ln and e a chapter of their own, covering integration, differentiation and exponential growth and decay.

Equations with the unknown in the exponent are the reason to learn any of this. To solve 3^x = 100, take logs of both sides so that x × log(3) = log(100), which gives x = 2/0.477, or 4.19. Equations with logs on both sides work in reverse: log(x) + log(x − 3) = 1 combines to log(x(x − 3)) = 1, so x(x − 3) = 10 and x² − 3x − 10 = 0, giving x = 5 once you reject x = −2, since you cannot take the log of a negative number. Finance uses the same move, because ln(future/present) ÷ ln(1+r) returns the number of years an investment needs to grow. GCSE covers the basic cases and A-Level takes in the rest.