worldofcalcsLogarithm Calculator
log₁₀ x
2.5378
log₁₀ 345 = ln 345 ÷ ln 10 = 5.8435 ÷ 2.3026 = 2.5378
log₁₀ x
2.5378
ln x
5.8435
log₂ x
8.4305
log 345 = 2.5378 · Characteristic: 2 · Mantissa: 0.5378
Antilog
Antilog: base raised to y
10^2.5378 = 344.9848
What is a logarithm?
A logarithm answers the question 'to what power must the base be raised to get this number?'. log₁₀ 1000 = 3 because 10³ = 1000. Common logs use base 10, natural logs (ln) use e ≈ 2.71828, and binary logs use base 2. This calculator works to any base, splits common logs into characteristic and mantissa as in Class 11 log tables, and finds antilogs.
Logarithm rules
log_b x = y ⇔ bʸ = x · log(xy) = log x + log y · log(x/y) = log x − log y · log(xⁿ) = n log x · log_b x = ln x ÷ ln b
The last rule, change of base, lets you find a logarithm to any base from natural or common logs. The logarithm is only defined for positive numbers, and the base must be positive and not 1.
Good to know
- The characteristic is one less than the number of digits before the decimal point for numbers of 1 or more.
- Numbers with the same digits, such as 345, 3.45 and 0.0345, share the same mantissa; only the characteristic changes.
- log 2 ≈ 0.3010 and log 3 ≈ 0.4771 are worth remembering: log 6 = log 2 + log 3 ≈ 0.7782.
- ln x ≈ 2.3026 × log₁₀ x.
- Logarithmic scales measure things that vary hugely: the Richter scale for earthquakes, decibels for sound and pH for acidity.
Worked example: log 345 using the log table method
- 345 = 3.45 × 10², so the characteristic is 2
- From the table (or the calculator), log 3.45 = 0.5378, the mantissa
- log 345 = 2 + 0.5378 = 2.5378
- For 0.0345 = 3.45 × 10⁻², the characteristic is −2 (written 2̄), so log 0.0345 = 2̄.5378 = −1.4622
- Antilog of 2.5378 = 10^2.5378 ≈ 345
Logarithms of 1 to 10
Common, natural and binary logs.
| x | log₁₀ x | ln x | log₂ x |
|---|---|---|---|
| 1 | 0.0000 | 0.0000 | 0.0000 |
| 2 | 0.3010 | 0.6931 | 1.0000 |
| 3 | 0.4771 | 1.0986 | 1.5850 |
| 4 | 0.6021 | 1.3863 | 2.0000 |
| 5 | 0.6990 | 1.6094 | 2.3219 |
| 6 | 0.7782 | 1.7918 | 2.5850 |
| 7 | 0.8451 | 1.9459 | 2.8074 |
| 8 | 0.9031 | 2.0794 | 3.0000 |
| 9 | 0.9542 | 2.1972 | 3.1699 |
| 10 | 1.0000 | 2.3026 | 3.3219 |
Rounded to 4 decimal places.