The rule
logₐ(x) = the power you raise a to in order to get x
Why it works
A logarithm answers the question 'to what power?'. If 10³ = 1000, then log₁₀(1000) = 3. It is the inverse of exponentiation, and it turns multiplication into addition, which is why it was invented.
How to do it by hand
- Ask what power the base needs to reach the number
- For any base, use the change of base formula: ln(x) ÷ ln(base)
- Check by raising the base to your answer
- The result should be the number you started with
What is worth knowing
Logarithms were invented by Napier in 1614 to turn multiplication into addition, and for three centuries slide rules and log tables were how serious arithmetic got done. That use is gone, but the mathematics is not: logarithmic scales handle quantities spanning many orders of magnitude, which is why decibels, pH, the Richter scale and stellar magnitudes are all logarithmic. Each is a case where the interesting thing is the ratio, not the difference. The three standard bases have three domains: 10 for engineering and measurement, e for calculus, and 2 for computer science, where it counts the doublings.