Natural Logarithm Calculator
The natural logarithm (ln) is the logarithm with base e, where e is Euler's number, approximately 2.718281828. The natural logarithm answers the question: what power must e be raised to in order to get this number? For example, ln(e) = 1 because e^1 = e, and ln(1) = 0 because e^0 = 1. The natural logarithm is fundamental in mathematics, appearing in calculus, differential equations, probability, and many areas of physics and engineering. It is preferred over log base 10 in advanced mathematics because its derivative and integral have simpler forms. This calculator computes the natural logarithm of any positive number and shows the verification: e^(ln(x)) = x. The relationship between the natural logarithm and the exponential function e^x makes them powerful tools for modeling continuous growth and decay.
Natural logarithm formula
ln(x) = log_e(x), where e ≈ 2.718281828
If e^x = n, then ln(n) = x
ln(1) = 0
ln(e) = 1
Verification: e^(ln(x)) = x
Natural logarithm reference values
| Number | Natural log (ln) | Number | Natural log (ln) |
|---|---|---|---|
| 0.5 | -0.69 | e | 1.00 |
| 1 | 0.00 | e^2 | 2.00 |
| 2 | 0.69 | 10 | 2.30 |
| 3 | 1.10 | 100 | 4.61 |
| 2.718 | 1.00 | 1,000 | 6.91 |
Natural logarithm calculator: frequently asked questions
What is the natural logarithm?
The natural logarithm (ln) is the logarithm with base e, where e is approximately 2.718281828. It is written as ln(x) or log_e(x). If e^x = n, then ln(n) = x. For example, e^1 = e, so ln(e) = 1.
What is e?
e is Euler's number, a mathematical constant approximately equal to 2.718281828. It appears frequently in mathematics, particularly in calculus, exponential growth, and logarithms. It is defined as the limit of (1 + 1/n)^n as n approaches infinity.
How is the natural logarithm different from log base 10?
The natural logarithm uses e (approximately 2.718) as the base, while log base 10 uses 10 as the base. The natural logarithm is more common in advanced mathematics, physics, and engineering because it has simpler derivatives and integrals.
What is e^x?
e^x is the exponential function, where e is raised to the power x. The natural logarithm and exponential function are inverses: ln(e^x) = x and e^(ln(x)) = x. The exponential function is widely used in modeling continuous growth.
Can you take the natural logarithm of a negative number?
No, the natural logarithm is undefined for zero and negative numbers in the real number system. It is only defined for positive numbers greater than zero.
Official sources
- Natural logarithm: NIST Special Publication 330.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.