How Logarithms Work
Logarithms are exponents in reverse. Where 10³ = 1000 asks “what is 10 cubed?”, log₁₀(1000) = 3 asks “10 to what power gives 1000?” Every exponential question has a mirror-image log question.
The Three Log Laws
- Product: log(xy) = log x + log y
- Quotient: log(x/y) = log x − log y
- Power: log(x^n) = n · log x
Domain Rules
You can only take logs of positive numbers, and the base must be positive and not 1. That is why log(−5) and log₁(x) are undefined — no power of a positive base gives a negative number.
Logs undo exponentials: try the exponent calculator or compute powers directly on the scientific calculator.