Logarithm Calculator

A logarithm answers the question: to what power must a given base be raised to produce a particular number? For example, log base 10 of 1000 = 3 because 10^3 = 1000. This calculator handles common logarithms (base 10), natural logarithms (base e, written as ln) and custom bases.

Logarithms are the inverse of exponents. They are used in science (pH scale, Richter scale, decibels), finance (compound interest) and computing (algorithmic complexity). The calculator shows the conversion between different bases using the change-of-base formula: log_b(x) = ln(x) / ln(b).

What is a logarithm? Inverse of exponentiation. log_b(x) = y means b^y = x. Sample: log₁₀(100) = 2 because 10² = 100. log₂(8) = 3 because 2³ = 8. Common logs: base 10 (log₁₀, often written 'log') and natural log (ln, base e ≈ 2.718). 'log' on UK calculators usually means log₁₀. UK A-Level Maths uses both extensively.

Laws of logarithms — essential identities. log(xy) = log(x) + log(y) — multiplication becomes addition. log(x/y) = log(x) − log(y) — division becomes subtraction. log(x^n) = n × log(x) — powers come down. log(1) = 0 (any base). log_b(b) = 1. log_b(x) = log(x) / log(b) — change of base formula (lets you compute any log using base 10 or e on calculator).

Why logarithms matter — real-world uses. Earthquakes: Richter scale logarithmic — magnitude 7 quake 10× larger than magnitude 6. Sound: decibels logarithmic — 10 dB increase = 10× louder. pH: logarithmic — pH 5 is 10× more acidic than pH 6. Star brightness: magnitude scale logarithmic. Information theory: bits = log₂(possibilities). Compound interest: ln(future/present) ÷ ln(1+r) = years to grow.

Natural log (ln) and e. Base e ≈ 2.71828... — Euler's number. Arises naturally in compound growth, calculus. d/dx (e^x) = e^x — unique property. d/dx (ln x) = 1/x. Continuous compound interest: A = Pe^(rt). UK A-Level Maths chapter heavily uses ln + e: integration, differentiation, exponential growth/decay. Used in: radioactive decay, population growth, Newton's law of cooling.

Solving logarithmic equations. Sample: 3^x = 100. Take log of both sides: x × log(3) = log(100) → x = 2/0.477 = 4.19. Sample: log(x) + log(x − 3) = 1. Combine: log(x(x − 3)) = 1, so x(x − 3) = 10, giving x² − 3x − 10 = 0, x = 5 (reject x = −2, can't take log of negative). UK GCSE: basic log equations. UK A-Level: full logarithmic equations including ln-based, change of base, log differentiation.