Natural Log Calculator - Calculate ln(x) Instantly
Compute the natural logarithm ln(x) of any positive number in one click — see the result, the formula, and worked examples side by side.
Enter a positive number and click Calculate to find its natural logarithm, the logarithm to the base of Euler's number e ≈ 2.71828.
Natural Log Calculator - Calculate ln(x) Instantly
Compute the natural logarithm ln(x) of any positive number in one click — see the result, the formula, and worked examples side by side.
The number must be greater than zero.
About the natural log calculator
The natural logarithm, written ln(x) or logₑ(x), is the logarithm whose base is Euler's number e ≈ 2.71828182845904523536. While common logarithms use base 10 and binary logarithms use base 2, the natural logarithm is the logarithm that appears most organically in calculus, differential equations, complex analysis, probability theory, and virtually every branch of quantitative science. Its name reflects this privileged status: among all logarithmic bases, only e produces the clean identity d/dx[ln x] = 1/x.
The definition is straightforward: ln(x) is the unique real number y such that e^y = x. Because e^0 = 1, we have ln(1) = 0. Because e^1 = e, we have ln(e) = 1. For any x > 1 the natural log is positive; for 0 < x < 1 it is negative; and as x approaches 0 from the right, ln(x) approaches negative infinity. The function is undefined for zero and for negative numbers in the real-number domain.
The natural log obeys the same algebraic laws as all logarithms. The product rule says ln(ab) = ln(a) + ln(b), which historically reduced multiplication to addition and made logarithm tables so valuable before electronic calculators. The quotient rule says ln(a/b) = ln(a) − ln(b). The power rule says ln(a^n) = n·ln(a), a fact exploited constantly when differentiating or integrating expressions involving exponentials.
In practice, the natural logarithm appears in an enormous range of applications. In finance, the continuously compounded return on an investment from price P₀ to price P₁ is ln(P₁/P₀). In information theory, the Shannon entropy of a probability distribution is computed with natural logs (giving units of nats rather than bits). In physics, the Boltzmann entropy formula S = k·ln(W) uses the natural log to connect microscopic disorder to macroscopic thermodynamic entropy. In chemistry, the Arrhenius equation and many rate-law expressions are linearised by taking the natural log of both sides. In statistics, the log-likelihood of a statistical model is almost always expressed in natural logs.
The natural log is also central to the study of prime numbers. The prime number theorem states that the number of primes up to N is approximately N/ln(N), one of the deepest results in analytic number theory. Calculus students encounter ln(x) as the antiderivative of 1/x, the one power function whose integral is not simply another power function.
This calculator evaluates ln(x) to ten significant digits using the IEEE-754 double-precision Math.log function, which is accurate to within one unit in the last place for all positive finite inputs. Enter any positive real number and the result, the symbolic equation, and the underlying formula are shown immediately.
Natural log examples
Four worked examples showing key values of the natural logarithm.
| Input (x) | ln(x) | Explanation |
|---|---|---|
| x = 1 | 0 | e⁰ = 1, so ln(1) = 0. The natural log of 1 is always zero regardless of the base. |
| x = e ≈ 2.71828 | 1 | By definition, e¹ = e, so ln(e) = 1. This is the fundamental identity of the natural logarithm. |
| x = 1000 | ≈ 6.9078 | ln(1000) = ln(10³) = 3·ln(10) ≈ 3 × 2.3026 = 6.9078. Demonstrates the power rule. |
| x = 0.5 | ≈ −0.6931 | For 0 < x < 1 the natural log is negative. ln(0.5) = ln(1/2) = −ln(2) ≈ −0.6931. |
How to use the natural log calculator
- Type any positive real number into the Number (x) field. The value must be strictly greater than zero.
- Click Calculate ln. The result appears instantly, showing the symbolic equation ln(x) = value alongside the formula.
- Click Reset to clear the input and start a new calculation.
- Use the example buttons under the worked-examples table to load common values — ln(1), ln(e), or ln(1000) — directly into the calculator.
- To compute the natural log of very small numbers such as 0.00001, type them in decimal form or scientific notation (e.g. 1e-5).
Natural log calculator FAQ
What is the natural logarithm?
The natural logarithm ln(x) is the logarithm to base e, Euler's number (≈ 2.71828). It answers the question: to what power must e be raised to get x? For example, ln(e²) = 2 because e raised to the power 2 equals e². It is the most important logarithm in mathematics and science because its derivative, d/dx[ln x] = 1/x, is cleaner than that of any other base.
Why is ln(1) = 0?
Because e⁰ = 1 for any base b, b⁰ = 1. Since ln asks for the exponent that makes e equal to x, and e⁰ = 1, it follows that ln(1) = 0. This is true for logarithms of any base: log_b(1) = 0 for all valid bases b > 0, b ≠ 1.
Can I take the natural log of a negative number?
Not in the real-number system. The exponential e^y is always positive for any real y, so there is no real exponent that makes e^y equal to a negative number. In the complex-number domain, ln of a negative number is defined (e.g. ln(−1) = iπ), but this calculator handles only real positive inputs.
How does ln relate to the common log (log₁₀)?
Both are logarithms, but with different bases. You can convert between them using the change-of-base formula: log₁₀(x) = ln(x) / ln(10) ≈ ln(x) / 2.302585. Conversely, ln(x) = log₁₀(x) × ln(10). The two functions differ by a constant multiplicative factor, so their graphs are vertical rescalings of each other.
What is ln(e^n) equal to?
By the inverse-function property, ln(e^n) = n for any real n. Taking the natural log is exactly the inverse operation of raising e to a power. This identity is used constantly in calculus and algebra to simplify expressions: if you have e^(2x) and want to solve for x, take ln of both sides to get 2x, then divide by 2.
Where is the natural logarithm used in real life?
The natural log appears in continuous compound interest (A = Pe^(rt), so t = ln(A/P)/r), in the Richter earthquake scale, in radioactive decay half-life calculations, in the measurement of musical intervals, and in machine learning (cross-entropy loss functions). Any time growth or decay is modelled by an exponential, the natural log is the tool that undoes or analyses it.