Negative Log Calculator (−log x)
Calculate −log₁₀(x) and −ln(x) for pH, pKa, pOH, and general use. Includes pH scale interpretation, concentration ↔ pH conversion, and reference tables.
Natural Log (ln) Calculator
Calculate the natural logarithm ln(x) = logₑ(x). See ln value, e^ln(x) verification, derivative 1/x, integral, and a full reference table of common ln values.
Presets
Results
Expression⧉
ln(10.00)
Logarithmic identity applied
ln(x)⧉
2.3025850930
x = 10.00000000
Verification e^(ln x)⧉
10.0000000000
Should equal x
Derivative 1/x⧉
0.1000000000
Slope of ln at this point
∫₁ˣ ln(t) dt⧉
14.02585093
Definite integral from 1 to x
log₁₀(x)⧉
1.0000000000
Common logarithm for comparison
log₂(x)⧉
3.3219280949
Binary logarithm
ln Magnitude Bars
0.01
-4.60517
0.10
-2.30259
0.25
-1.38629
0.50
-0.69315
1.00
0.00000
e
1.00000
2.00
0.69315
3.00
1.09861
5.00
1.60944
7.00
1.94591
10.00
2.30259
20.00
2.99573
50.00
3.91202
100.00
4.60517
500.00
6.21461
1,000.00
6.90776
Natural Log Reference Table
| x | ln(x) | e^(ln x) | 1/x |
|---|---|---|---|
| 0.01 | -4.605170 | 0.010000 | 100.000000 |
| 0.10 | -2.302585 | 0.100000 | 10.000000 |
| 0.25 | -1.386294 | 0.250000 | 4.000000 |
| 0.50 | -0.693147 | 0.500000 | 2.000000 |
| 1.00 | 0.000000 | 1.000000 | 1.000000 |
| e ≈ 2.718 | 1.000000 | 2.718282 | 0.367879 |
| 2.00 | 0.693147 | 2.000000 | 0.500000 |
| 3.00 | 1.098612 | 3.000000 | 0.333333 |
| 5.00 | 1.609438 | 5.000000 | 0.200000 |
| 7.00 | 1.945910 | 7.000000 | 0.142857 |
| 10.00 | 2.302585 | 10.000000 | 0.100000 |
| 20.00 | 2.995732 | 20.000000 | 0.050000 |
| 50.00 | 3.912023 | 50.000000 | 0.020000 |
| 100.00 | 4.605170 | 100.000000 | 0.010000 |
| 500.00 | 6.214608 | 500.000000 | 0.002000 |
| 1,000.00 | 6.907755 | 1,000.000000 | 0.001000 |
Planning notes, formulas, and examples
About the Natural Log (ln) Calculator
The natural logarithm, written ln(x) or logₑ(x), is the logarithm to the base e ≈ 2.71828. It is one of the most important functions in mathematics, appearing throughout calculus, differential equations, probability, physics, and engineering. The natural log answers the question: "To what power must e be raised to get x?" This calculator lets you enter any positive number and see ln(x), along with the verification eˡⁿ⁽ˣ⁾ = x, the derivative value 1/x at that point, and the definite integral of ln from 1 to x. A comprehensive reference table lists ln values for common numbers from 0.01 to 1000, and the magnitude bar chart gives you a visual sense of how ln grows logarithmically. Use the presets to jump to important values like e, e², 10, 100, or 1/e without typing. Whether you are studying calculus, solving exponential equations, or working with growth and decay models, the page gives you the surrounding relationships as well as the log value itself.
When This Page Helps
Natural-log questions usually sit inside broader calculus or exponential-model work, so seeing only ln(x) is rarely enough. This page is useful because it keeps the log value, the exponential verification, the derivative, and the integral context together, making it easier to check whether the logarithm fits the rest of the problem.
How to Use the Inputs
- Enter Value b and Exponent n in the input fields.
- Select the mode, method, or precision options that match your natural log (ln) problem.
- Read Expression first, then use ln(x) to confirm your setup is correct.
- Try a preset such as "ln(x)" to test a known case quickly.
Formula used
ln(x) = logₑ(x) where e ≈ 2.71828. Key identities: ln(1) = 0, ln(e) = 1, ln(ab) = ln(a) + ln(b), ln(a/b) = ln(a) − ln(b), ln(aⁿ) = n·ln(a). Derivative: d/dx ln(x) = 1/x. Integral: ∫₁ˣ ln(t) dt = x·ln(x) − x + 1.Example Calculation
Result: Expression shown by the calculator
Using the preset "ln(x)", the calculator evaluates the natural log (ln) setup, applies the selected algebra rules, and reports Expression with supporting checks so you can verify each transformation.
Tips & Best Practices
- ln(x) is only defined for x > 0. For x ≤ 0 you enter the complex domain.
- ln(1) = 0 and ln(e) = 1 are the two most fundamental reference points.
- Use the change-of-base formula log_b(x) = ln(x) / ln(b) to convert any logarithm to natural log.
- ln grows very slowly — ln(1 000 000) is only about 13.8.
How This Natural Log (ln) Calculator Works
This calculator takes Value b, Exponent n and applies the relevant natural log (ln) relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Interpreting Results
Start with the primary output, then use Expression, ln(x), Verification e^(ln x), Derivative 1/x to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
Study Strategy
Sources & Methodology
Last updated:
Frequently Asked Questions
The natural logarithm ln(x) is the inverse of the exponential function eˣ. It tells you the exponent to which e must be raised to produce x.
e is the unique base for which the derivative of the exponential function equals itself: d/dx eˣ = eˣ. This makes e the natural choice for calculus.
ln(0) is undefined (it approaches −∞). The natural log is only defined for positive real numbers.
ln(x) = log₁₀(x) / log₁₀(e) ≈ log₁₀(x) × 2.302585. Conversely, log₁₀(x) = ln(x) / ln(10).
∫ ln(x) dx = x·ln(x) − x + C, found via integration by parts.
d/dx ln(x) = 1/x for x > 0.
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