Logarithm Rules
| Rule | Formula | Example |
|---|---|---|
| Product | log(ab) = log(a) + log(b) | log(50) = log(5) + log(10) = 0.699 + 1 = 1.699 |
| Quotient | log(a/b) = log(a) − log(b) | log(5) = log(50) − log(10) = 1.699 − 1 = 0.699 |
| Power | log(an) = n·log(a) | log(8) = 3·log(2) = 3 × 0.301 = 0.903 |
| Change of Base | logb(x) = ln(x)/ln(b) | log₅(125) = ln(125)/ln(5) = 4.828/1.609 = 3 |
| Log of 1 | logb(1) = 0 | Any base: log(1) = 0 |
| Log of Base | logb(b) = 1 | log₁₀(10) = 1, ln(e) = 1 |
Common Logarithm Values
| x | log₁₀(x) | ln(x) | log₂(x) |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.3010 | 0.6931 | 1 |
| e | 0.4343 | 1 | 1.4427 |
| 10 | 1 | 2.3026 | 3.3219 |
| 100 | 2 | 4.6052 | 6.6439 |
| 1000 | 3 | 6.9078 | 9.9658 |