Logarithm Calculator
A logarithm answers the question: "To what power must the base be raised to get this number?" log_b(x) = n means bⁿ = x. Logarithms compress enormous ranges and are used in sound (decibels), earthquakes (Richter scale), and pH.
- 1log_b(x) = n means bⁿ = x
- 2Common log (log₁₀): log(1000) = 3 because 10³ = 1000
- 3Natural log (ln): ln(e) = 1 because e¹ = e ≈ 2.718
- 4log_b(x) = log(x)/log(b) — change of base formula
log₁₀(100)=210² = 100
log₂(32)=52⁵ = 32
ln(e³)=3Natural log of e³
log₅(125)=35³ = 125
| Rule | Formula |
|---|---|
| Product | log(xy) = log(x) + log(y) |
| Quotient | log(x/y) = log(x) − log(y) |
| Power | log(xⁿ) = n·log(x) |
| Change of base | log_b(x) = ln(x)/ln(b) |
| Inverse | b^(log_b(x)) = x |
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Fun Fact
A 9.0 earthquake releases about 31 times the energy of an 8.0, because the Richter scale is logarithmic (base 10). Each whole number represents 10× more ground motion.
References
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