Logarithm Calculator
A logarithm is the inverse of exponentiation. log_b(x) answers "to what power must b be raised to get x?" Common logarithm (base 10) is used in science; natural logarithm (base e) in calculus; binary logarithm (base 2) in computer science.
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Tip: To estimate log₁₀ of any number mentally: count the digits and subtract 1. log₁₀(8,432) ≈ 3.9 (4 digits → between 3 and 4).
- 1log_b(x) = n means b^n = x
- 2Common log: log₁₀(1000) = 3 because 10³ = 1000
- 3Natural log: ln(e) = 1 because e^1 = e
- 4Change of base: log_b(x) = log(x) / log(b)
log₁₀(10,000)=410⁴ = 10,000
log₂(256)=82⁸ = 256
ln(20.09)=≈ 3e³ ≈ 20.09
| Rule | Formula | Example |
|---|---|---|
| Product | log(xy) = log(x) + log(y) | log(100×10) = log(100)+log(10) = 3 |
| Quotient | log(x/y) = log(x) − log(y) | log(100/10) = 2−1 = 1 |
| Power | log(xⁿ) = n·log(x) | log(1000²) = 2×3 = 6 |
| Change of base | log_b(x) = ln(x)/ln(b) | log₂(8) = ln8/ln2 = 3 |
| Inverse | b^(log_b(x)) = x | 10^(log₁₀(50)) = 50 |
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Fun Fact
The Richter scale, pH scale, decibel scale, and musical octaves are all logarithmic. A pH difference of 1 represents a 10× change in acidity, not a 1-unit linear change.
References
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