Natural Log Calculator (ln)
Calculate ln(x) instantly. Shows ln alongside log₁₀ and log₂ for comparison. Switch to Antilog for eˣ, or use Exponent and Scientific Notation tabs for related calculations.
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How It Works
The calculator opens with base e selected. Enter any positive number and press Calculate. The result shows ln(x) as the primary value, with log₁₀ and log₂ alongside. The verify line confirms: e^result ≈ your input. For very large inputs (e.g. ln(1e100) = 230.26), the result stays accurate because the formula is math.log(x) directly.
ln(x) = logₑ(x) | ln(e) = 1 | ln(1) = 0 | ln(eⁿ) = nln(1000) = 6.9078 | ln(e²) = 2 | ln(0.5) = −0.6931Switch to the Antilog tab and ensure base e is selected. Enter the exponent x — the result is eˣ. Since eˣ and ln are inverses: e^ln(7) = 7 and ln(e^7) = 7 exactly. Used to recover the original value after taking the natural log, or to evaluate exponential growth e^(rt).
antilogₑ(x) = eˣ | e^ln(x) = x | e⁰ = 1 | e¹ = 2.71828e² = 7.38906 | e^0.5 = 1.64872 | e^(−1) = 0.36788Three rules let you break complex natural logs into simple sums. Product: ln(xy) = ln(x) + ln(y). Quotient: ln(x/y) = ln(x) − ln(y). Power: ln(xⁿ) = n·ln(x). Verify any rule: ln(6) = ln(2×3) = ln(2) + ln(3) = 0.693 + 1.099 = 1.792 — match the direct result.
ln(xy) = ln(x) + ln(y) | ln(xⁿ) = n·ln(x)ln(8) = 3·ln(2) = 3×0.6931 = 2.0794The change-of-base formula: log_b(x) = ln(x) ÷ ln(b). The Log/Ln tab handles this automatically — type any base and the calculator divides ln(x) by ln(b) internally. Example: log₅(125) = ln(125) ÷ ln(5) = 4.8283 ÷ 1.6094 = 3. Useful when your base is not 2, e, or 10.
log_b(x) = ln(x) ÷ ln(b)log₅(125) = ln(125)/ln(5) = 4.828/1.609 = 3 ✓For continuous growth A = Pe^(rt): time to double = ln(2) ÷ r ≈ 0.6931 ÷ r. At 5% rate: 0.6931 ÷ 0.05 = 13.86 years. Growth rate from doubling time: r = ln(2) ÷ t. From A and P: r = ln(A/P) ÷ t. Enter ln(2) = 0.693147 directly — or use ln(A/P) for any growth multiple.
t = ln(A/P) ÷ r | ln(2) ≈ 0.6931 | ln(10) ≈ 2.3026Doubling at 7%: t = ln(2) ÷ 0.07 = 0.6931/0.07 ≈ 9.9 yearsQuick Reference
Key ln and eˣ values — verify these in the calculator above.
ln — base e
ln(1)
0
ln — base e
ln(e)
1
ln — base e
ln(e²)
2
ln — base e
ln(10)
2.302585
ln — doubling
ln(2)
0.693147
Antilog e
e^1
2.71828
Antilog e
e^2
7.38906
Antilog e
e^(−1)
0.36788
Tips & Shortcuts
ln(x) and log₁₀(x) are shown simultaneously — no need to switch or recalculate. The conversion factor is ln(10) ≈ 2.3026: multiply log₁₀ by 2.3026 to get ln.
For eˣ calculations, use the Antilog tab with base e selected. This is faster than using the Exponent tab with base 2.71828 and avoids rounding errors in the base value.
ln(1) = 0 always, regardless of base. This is because e⁰ = 1. Similarly, ln(e) = 1 exactly. Use these as quick sanity checks after a calculation.
To solve exponential equations like e^(2t) = 15: take ln of both sides → 2t = ln(15) ≈ 2.708 → t ≈ 1.354. Enter 15 in the Log/Ln tab with base e to get ln(15) directly.
The verify line (e^result = x) confirms accuracy — if the verify output does not match your input, the precision of the display is limited by floating-point, not a calculation error.
Common Mistakes
Entering x = 0 or a negative number into the ln calculation
ln is only defined for x > 0. ln(0) = −∞ and ln(negative) is undefined in real numbers. The calculator returns an error — check that your input is strictly positive before calculating.
Confusing ln(x²) with (ln x)²
ln(x²) = 2·ln(x) by the power rule. But (ln x)² means squaring the result of ln. Example: ln(4) = 2·ln(2) ≈ 1.386. (ln 2)² ≈ 0.480. These are completely different — apply the power rule only when the exponent is inside the argument.
Using ln when the formula requires log₁₀ (or vice versa)
pH = −log₁₀[H⁺] uses base 10, not ln. Continuous compound interest A=Pe^(rt) uses ln to solve for t. Always check the base used in the formula — in engineering "log" means base 10; in calculus it often means ln.
Calculating ln(a + b) as ln(a) + ln(b)
The product rule is ln(a × b) = ln(a) + ln(b), NOT ln(a + b). There is no simplification for ln(a + b). Example: ln(2 + 3) = ln(5) ≈ 1.609, but ln(2) + ln(3) ≈ 0.693 + 1.099 = 1.792 — a different value.
Expecting ln(eˣ) to differ from x
ln and eˣ are perfect inverses: ln(eˣ) = x and e^ln(x) = x exactly. If you get a slightly different number, it is floating-point rounding in the final display digit, not a real difference. The verify line in the Antilog tab confirms this identity.
Frequently Asked Questions
ln(x) is the logarithm with base e ≈ 2.71828182845905. It answers: "to what power must e be raised to equal x?" ln(e) = 1, ln(e²) = 2, ln(1) = 0. It is the inverse of eˣ — applying both returns the original value: ln(eˣ) = x and e^ln(x) = x.
ln uses base e. log (or log₁₀) uses base 10. For the same input: ln(100) ≈ 4.605, log₁₀(100) = 2. They differ by the factor ln(10) ≈ 2.3026. In physics and pure math, "log" often means ln. In engineering and chemistry, "log" means log₁₀. Always check the context or the formula documentation.
eˣ is the inverse (antilog base e) of ln. Switch to the Antilog tab, enter your exponent x, and make sure base e is selected. e¹ = 2.71828, e² = 7.38906, e^0 = 1, e^(−1) = 0.36788. Since ln and eˣ are inverses, e^ln(5) = 5 exactly.
Continuous compounding A = Pe^(rt) uses e because it is the limit of (1 + 1/n)ⁿ as n → ∞ — the most natural base for continuous processes. To find doubling time: t = ln(2) ÷ r. At 7% annual rate: t = 0.6931 ÷ 0.07 ≈ 9.9 years. This also explains the Rule of 70: doubling time ≈ 70 ÷ interest rate%.
ln(0) = −∞ — the natural log approaches negative infinity as x approaches zero from the right. For negative numbers, ln is undefined in real mathematics because no real power of e produces a negative result (eˣ is always positive). The calculator returns an error for x ≤ 0. In complex math, ln(−1) = iπ, but this is outside real-number scope.
Product rule: ln(xy) = ln(x) + ln(y). Example: ln(6) = ln(2) + ln(3) = 0.693 + 1.099 = 1.792. Quotient rule: ln(x/y) = ln(x) − ln(y). Power rule: ln(xⁿ) = n·ln(x). Example: ln(8) = ln(2³) = 3·ln(2) = 3 × 0.693 = 2.079. These rules reduce complex expressions to sums and multiples of simpler logs.
ln(x) = log₁₀(x) × 2.302585. log₁₀(x) = ln(x) × 0.434294. The calculator shows both simultaneously — no manual conversion needed. For example, entering x = 1000 shows ln = 6.9078 and log₁₀ = 3. Verify: 6.9078 × 0.434294 ≈ 3 ✓
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