The undo button for exponents.
The inverse function to exponentiation. log_b(x) asks 'b to what power equals x?'
Invented by John Napier in 1614 to help sailors perform massive multiplication problems by turning them into addition.
Log of a Product = Sum of Logs. Log(ab) = Log(a) + Log(b).
Plain English: A log is just a question: 'How many times do I multiply this number to get that number?' Log-base-10 of 1000 is 3, because 10x10x10 is 1000. It measures magnitude, not just quantity.
Real-world example: Earthquakes. The Richter Scale is logarithmic. A 6.0 is 10 times stronger than a 5.0, not just a little bit stronger.
Confusing the Base and the Argument. Remember the 'Snail' motion: Base to the Answer equals the Middle.