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 5/5
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 5/5
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 5/5 · 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 does this natural log (ln) calculator compute?

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.

The published formula or method is y = ln(x) ⇔ eʸ = x. The following references support the stated mathematics or method:

NIST DLMF §4.2 — Logarithm, Exponential, Powers: Definitions and mathematical identities for logarithms, exponentials, and powers.

Which inputs and limits should you check?

Follow the input domain, units, branch or solve direction, and rounding policy in the specification. Preserve exact outputs and any additional candidates or warnings.

The worked example uses Base definition with ln(e). Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.ln v1.0.4 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

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.

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?