Math calculator

Natural Log (ln) Calculator

Q: Can ln(x) be negative?

Yes. For 0 < x < 1, ln(x) is negative (e.g. ln(0.5) ≈ −0.693).

Q: How do I convert ln to log₁₀?

log₁₀(x) = ln(x) / ln(10) ≈ ln(x) / 2.302585.

Q: Where is ln used in real life?

Continuous compound interest, population growth/decay, radioactive half-life, capacitor discharge, cooling laws, and the rule-of-70 doubling-time estimate.

Q: What if x is 0 or negative?

Real ln is undefined for x ≤ 0. This calculator returns INVALID_ARGUMENT. As x → 0⁺, ln(x) → −∞.

Calculate ln(x) = log_e(x) for x > 0. Fast local calculation with shareable, machine-readable results.

Instant result

Result

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Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance

Input interpretation: Enter values to calculate.
Result: —
Assurance: Core
Declared partition coverage: PASS · 3/3 declared partitions (ln, invalid-domain, xcal) · Matrix
Numerical scope: ≤2 ULP vs O3 applies to the 16 published tabulated natural-log vectors. It is not a guarantee over the whole positive reals or unlisted neighborhoods of 0 and 1.
Known limitations: Real ln: x > 0
Core CVP does not include live graph, viewport, or pointer interaction.
Model: The real natural logarithm y = ln(x) = log_e(x) for x > 0.
Scope: Real arithmetic; argument x > 0. Base is e, not 10.
Verification: Engine tested · Source checked · v1.0.4 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions: Calculation 1.0.4 · CVP protocol 1.0.0-proposed · Evidence 2026-09-08.xcal
Verification revision: 2026-09-08.xcal · 2/2 property · digest eafd28643bc3
Legacy regression: 23/23 tests · Production surface contract 3/3
Reference: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP / URL→result→graph — Success 1/1. Integration: URL → SSR result → Live graph current point (4/4). Hydration/slider/history are URL-canonical contracts, not a live browser session.
Supplemental domain review: Not performed
Named expert review: Not performed
CVP suite: 3/3 golden · 12/12 CVP boundary · 12/12 invalid · 2/2 property · 2/2 metamorphic · 3/3 round-trip · 16/16 O3 · 1/1 cross-interface · 12/12 cross-calculator · 4/4 URL→result→graph · 1/1 CVP contract · Manifest
Sources: NIST Digital Library of Mathematical Functions, Chapter 4 — Logarithm, Exponential, Powers
NIST DLMF §4.2 — Logarithm: definitions and inverse of the exponential
CalculatorX mathematical conventions — Natural logarithm base e; real argument x > 0
Sources
Evidence: 11 legacy golden · 12 legacy boundary · legacy regression suite · 3/3 oracle-backed golden · 12/12 invalid · Artifact integrity PASS
Semantic contract: PASS

Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Definitionln(x) = y ⇔ eʸ = x

As log_eln(x) = log_e(x)

To common loglog₁₀(x) = ln(x) / ln(10)

Change of baselog_b(x) = ln(x) / ln(b)

iRequire x > 0. ln(1)=0 exactly. With the binary64 constant Math.E, ln(e) rounds to 1 (e itself is not exactly representable). ln(0) undefined. In some programming languages log(x) means ln(x); on many calculators log means log₁₀.

How to use

Enter the argument x

The published engine computes ln(x) for x > 0. Type e for Euler’s number. Inverse x = e^y is Antilog with base e; share URLs use x.

Read ln(x) and the reverse check

The main result is y = ln(x). The check line confirms e^y ≈ x. Explore x is a log-scale window 0.1–10, not the domain; type any x > 0.

Share the result

Share URLs include x, for example /calc/math/ln?x=10.

Example calculations

Common configurations with formula and result.

ln of e

Base definition

ln(e)

ln of 1

e⁰ = 1

ln(1)

ln of 2

Doubling / half-life constant

ln(2)

≈ 0.693147

ln of 10

Link to common log

ln(10)

≈ 2.302585

ln of 0.5

Negative result

ln(0.5)

≈ −0.693147

Common ln values

Common values at a glance.

x	ln(x)
1	0
2	≈ 0.693147
e ≈ 2.71828	1
3	≈ 1.098612
10	≈ 2.302585
20	≈ 2.995732
50	≈ 3.912023
100	≈ 4.605170

i ln 2 ≈ 0.6931 appears in doubling-time ≈ 70/r (%) and in radioactive half-life formulas.

Natural Log (ln) calculator specification

Version 1.0.4 · Engine tested

Calculation status

Engine tested 23/23 tests · Production surface contract 3/3
Named expert review Not performed
Calculation version 1.0.4

Review policy · Evidence

Definition: The natural logarithm ln(x) is the logarithm with base e ≈ 2.718281828. It answers “to what power must e be raised to get x?” If e^y = x, then ln(x) = y. The published engine computes y = ln(x) for x > 0. Inverse x = e^y is the Antilog calculator with base e.
What it calculates: The real natural logarithm y = ln(x) = log_e(x) for x > 0.
Inputs: Argument x > 0 (the token e is accepted as Euler’s number)
Outputs: y: natural logarithm ln(x)
verification: reverse check exp(result) against x
Formula: y = ln(x) ⇔ eʸ = x
Assumptions: Real arithmetic; argument x > 0. Base is e, not 10.
On many calculators log means log₁₀; in some programming languages log(x) means ln(x).
Published engine is y = ln(x). A legacy y query parameter is ignored and stripped from the address bar. Inverse x = e^y is Antilog with base e.
Units: Dimensionless numeric result. In physical applications, logarithms are normally applied to dimensionless ratios or normalized quantities.
Boundary conditions: Missing x returns MISSING_REQUIRED_INPUT.
x ≤ 0 returns INVALID_ARGUMENT (real ln is undefined).
x = 1 returns 0 (exact). The token e and the binary64 constant Math.E return 1 with the exact display flag; Math.E is a rounded approximation of the mathematical constant e, not an exact representation.
0 < x < 1 yields a negative result (for example ln(0.5) = −ln(2)).
For every finite x > 0 in binary64, y = ln(x) is finite (about −744 < y < 710). Domain errors use INVALID_ARGUMENT; the engine does not return Infinity for valid positive finite inputs.
Numerical precision: Computation uses IEEE-754 binary64 (JavaScript Number): y = Math.log(x).
REST and SSR return { y, value, verification }. value equals y so older scalar readers still work. The on-page result may round for display (up to 12 significant digits; scientific notation when 0 < |y| < 1e-6). Finite binary64 ln never reaches |y| ≥ 1e12.
ln(1) is exactly 0. ln(Math.E) rounds to 1 in binary64 and is marked exact for display; that does not mean the mathematical constant e is exactly representable. Irrational results such as ln(2) and ln(10) are rounded on the page; the API keeps the full binary64 value.
Interactive evaluation runs in the browser. Shared URLs and REST use the same engine server-side. Share URLs include x only, for example /calc/math/ln?x=10.
The Result Card includes an engine-linked Live graph of y = ln(x) using the same Overview / Focus template as Log (canonical window 0.1 ≤ x ≤ 10; domain x > 0). Explore x is an engine-linked log-scale slider on that same window: dragging it writes x and updates the result, reverse check, and graph through the same local engine. Values outside 0.1–10 (any x > 0) are typed in the argument field.
Example: ln(10) ≈ 2.302585092994
Validation cases: 10 published on this page · 23/23 tests · Production surface contract 3/3 · View evidence
x=1 → 0
x=e → 1
x=2 → ≈0.693147180560
x=10 → ≈2.30258509299
x=100 → ≈4.60517018599
x=0.5 → ≈−0.693147180560
x=3 → ≈1.09861228867
x=0 → error INVALID_ARGUMENT
x=−1 → error INVALID_ARGUMENT
empty x → error MISSING_REQUIRED_INPUT
Sources: NIST Digital Library of Mathematical Functions, Chapter 4 — Logarithm, Exponential, Powers
Supports: Natural logarithm as the inverse of the exponential; ln(x)=y ⇔ e^y=x
NIST DLMF §4.2 — Logarithm: definitions and inverse of the exponential
Supports: e^(ln x) = x and ln(e^y) = y for real x > 0
CalculatorX mathematical conventions — Natural logarithm base e; real argument x > 0
Supports: This page is ln only. Common log is the Log calculator; inverse is Antilog with base e.
Calculation version: 1.0.4

Background

Interpretation and common distinctions.

What is the natural logarithm?

The natural logarithm of a positive number x is the power to which Euler’s number e ≈ 2.718281828 must be raised to get x:

ln(x) = y Longleftrightarrow e^y = x

Also written logₑ(x). The name comes from Latin logarithmus naturalis.

Key values:

ln(1) = 0 because e^0 = 1
With the binary64 constant Math.E, ln(e) rounds to 1 (mathematical e itself is not exactly representable in binary64)
ln(2) ≈ 0.693147
ln(10) ≈ 2.302585

ln vs log₁₀

	Natural	Common
Symbol	ln(x)	log₁₀(x) or often log(x)
Base	e	10

log₁₀(x) = (ln(x))/(ln(10)) ≈ (ln(x))/2.302585

log_b(x) = (ln(x))/(ln(b))

Rules

Same product / quotient / power rules as other logs:

ln(xy)=ln x+ln y, ln(x/y)=ln x−ln y, ln(x^n)=nln x, ln(e^x)=x

Graph sketch

Plotting this calculator’s input x against result y:

y = ln(x)

As x → 0⁺, y → −∞
As x → +∞, y grows without bound
At x = 1, y = 0; at x = e, y = 1

Why ln 2 matters

Doubling time at continuous rate r% per period is roughly 70/r years (from 100ln 2 / r). Half-life formulas also use ln 2.

Where ln appears

Continuous compounding, exponential growth/decay, radioactive half-life, RC circuits, Newton’s law of cooling, and calculus (d/dx ln x = 1/x).

CVP (Calculator Verification Protocol)

Math pilot under CVP 1.0 Proposed. Profile: core (pure natural logarithm, not an engineering model).

Evidence Manifest: /evidence/math.ln/1.0.4.cvp.json
Independent reference: O1 (model) + O3 mpmath/MPFR expected values and numerical behavior + O2 live identities
Numerical policy: IEEE-754 binary64; ≤2 ULP vs O3 applies to the 16 published tabulated natural-log vectors. It is not a guarantee over the whole positive reals or unlisted neighborhoods of 0 and 1.declared_before_evaluation=true
Interfaces: UI (SSR), REST, and MCP must agree. A separate integration check covers URL → result → Live graph. ui-ssr is not a live browser session.
Cross-calculator (XCAL): antilog(ln x, e) ≈ x, log_e(x) = ln(x), and ln(e^n) ≈ n vs Antilog / Log / Exponent. Exponential growth is related-tool N/A.
Expert review is optional and is not required for CVP Verified.

Other calculators in this family: Antilog Calculator, Exponent Calculator, Exponential Growth Calculator, Log Calculator, Root Calculator, Square Root Calculator . Explore all Powers, Roots & Logarithms.

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Frequently asked questions

Key distinctions behind the calculation.

What is ln(x)?

The natural logarithm: the exponent y such that e^y = x. Written ln(x) or log_e(x).

What is the difference between ln and log?

ln uses base e. On many scientific calculators, log means log₁₀. In some programming languages, log(x) means ln(x).

Why is it called natural?

Because it is the inverse of e^x, the unique exponential whose derivative equals itself. That makes ln and e^x central in calculus and continuous growth/decay models.

What is ln(1)?

0, because e⁰ = 1.

What is ln(0)?

Undefined for real numbers. e^y is never zero; as x → 0⁺, ln(x) → −∞.

Can ln(x) be negative?