Logarithm Calculator

Pro tips most people learn the hard way

• Each +1 on a log scale is a 10x multiplier (in base 10). Richter 7 is 10x the ground motion of Richter 6. pH 4 is 1,000x more acidic than pH 7 (three units, 10³). This is why small numbers on log scales mean massive real-world differences — your intuition for 'the difference between 6 and 7' is way off.
• Pre-calculator engineers used log tables to multiply huge numbers by adding their logs together. Napier invented this in 1614 and it was the labor-saving math hack of its century. Slide rules implement the same trick physically — sliding two log scales adds the logs, which multiplies the original numbers. Mechanical multiplication via geometry.
• Binary log (log2) is the CS workhorse. log2(n) is the number of bits needed to represent n distinct values, and it's the complexity of binary search and balanced binary trees. Why search engines can return results in milliseconds across billions of documents: log2(1 billion) is only about 30, meaning 30 comparisons gets you there.
• Natural log (base e) is what calculus prefers, because d/dx[ln(x)] = 1/x is the cleanest derivative you'll ever see. e itself is the 'natural' base for continuous growth — anything growing exponentially in continuous time is naturally expressed with e. Compound interest with infinitely small compounding intervals lands on e.
• Honest take: log scales fool people because they look like linear scales unless you're paying attention. A graph showing 'magnitude 1 to 9' looks like a steady gradient, but the actual data spans 9 orders of magnitude (a factor of a billion). Always check whether a chart's axes are linear or log — it changes the interpretation completely.