Logarithm Explorer

Logarithms turn multiplication into addition — that's the whole trick, and it's why they show up everywhere humans need to compress huge ranges into something readable. The pH scale is -log₁₀[H⁺]: each whole step is a 10× change in acidity. Decibels are 10·log₁₀(power ratio): a quiet room and a jet engine differ by roughly a trillion in raw power, but only by about 120 dB. The Richter/moment magnitude scale works the same way — a magnitude 7 earthquake releases about 32 times more energy than a magnitude 6, not 7/6 times more. Before calculators, engineers carried slide rules that turned this addition trick into a physical ruler: sliding two logarithmically-marked scales let you multiply two numbers just by adding lengths. And in information theory, the number of bits needed to represent something is literally log₂ of the number of possibilities — every doubling of choices costs exactly one more bit.

A logarithm asks a backwards question: b to what power gives x? That answer is the logarithm — log_b(x) is simply the exponent. That is why every curve in this tool passes through (1, 0): any base raised to the power 0 equals 1. The two references shown, ln(x) and log10(x), are the same question asked with base e and base 10.

Slide the base and the graphs seem to change shape — but they do not. Change of base gives log_b(x) = ln(x) / ln(b), so each curve is just ln(x) multiplied by a fixed number. The slider only stretches one universal curve vertically. Push the base below 1 and that factor turns negative, flipping the curve upside down; set it exactly to 1 and everything collapses, since ln(1) = 0 means dividing by zero.