Logarithm Calculator
The logarithm of a number to any base — ln, log and log₂: exact when the number is a power of the base, with the check and the graph
Logarithm Calculator
Natural, Common and Binary Logarithms
logb x is the power you raise b to, to get x: by = x; b > 0, b ≠ 1, x > 0.
ln is the natural log, base e ≈ 2.71828; log is the common log, base 10; log₂ is the binary log, base 2.
| x | ln x | log x | log₂ x |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | ≈ 0.6931 | ≈ 0.3010 | 1 |
| 3 | ≈ 1.0986 | ≈ 0.4771 | ≈ 1.5850 |
| 4 | ≈ 1.3863 | ≈ 0.6021 | 2 |
| 5 | ≈ 1.6094 | ≈ 0.6990 | ≈ 2.3219 |
| 6 | ≈ 1.7918 | ≈ 0.7782 | ≈ 2.5850 |
| 7 | ≈ 1.9459 | ≈ 0.8451 | ≈ 2.8074 |
| 8 | ≈ 2.0794 | ≈ 0.9031 | 3 |
| 9 | ≈ 2.1972 | ≈ 0.9542 | ≈ 3.1699 |
| 10 | ≈ 2.3026 | 1 | ≈ 3.3219 |
Example: ln 2
ln 2 ≈ 0.6931
ln 2 is the power you raise e ≈ 2.71828 to, to get 2: ey = 2.
e⁰ = 1 < 2 < e¹ ≈ 2.7183, so 0 < ln 2 < 1
Answer point ≈ (2, 0.6931).
The curve y = ln x passes through (1, 0) and (e, 1) and rises: for x > 1 the log is positive, for 0 < x < 1 negative.
Common mistakes
- log without a base: on a calculator and at school it is base 10 (log 100 = 2), while Python's math.log and Excel's LN use base e (ln 100 ≈ 4.6052); Excel's LOG is base 10 again.
- ln 0 and the log of a negative number do not exist: e raised to any power is positive.
- There is no base 1: 1 raised to any power is 1.
See Also
- Square Root Calculator - Calculate the square root of any number
- Simplify Square Root Calculator - Write any square root in simplest radical form, step by step
- Scientific Calculator with Memory & History - Free Engineering Calculator Online with Memory Function, Operation History, Logarithms & Trigonometry. Similar to ti-83, ti-84, ti-89
FAQ
What is the difference between ln, log and lg?
ln is the natural log, base e ≈ 2.71828; log on a calculator is the common log, base 10 (some textbooks write it lg). They differ by a constant factor: log x = ln x / ln 10, so ln 2 ≈ 0.6931 and log 2 ≈ 0.3010.
How to calculate log base 2 on a calculator?
Divide two natural logs: log₂ 5 = ln 5 / ln 2 ≈ 1.609438 / 0.693147 ≈ 2.3219. Any base works the same way: log_b x = ln x / ln b.
What is ln of infinity?
ln x has no maximum: it grows without bound as x grows, only slowly — ln 1,000,000,000 ≈ 20.7233. So ln ∞ is not a number; one says that ln x tends to infinity.