Natural Log Calculator
The natural logarithm, ln(x), answers the question “to what power must e be raised to get x?” It is the logarithm with base e, about 2.71828, and it sits at the centre of calculus, continuous growth and decay. This calculator finds ln(x) and, because one base is rarely enough, also returns log base 10, log base 2 and any custom base in the same run, then reverses the operation entirely in antilog mode.
Every result carries its own audit trail: the change-of-base steps that produced it, a verification row that raises the base back to the answer, and the ln curve with your point marked on it. Enter any positive number and the full picture arrives in one click.
Real logarithms are only defined for positive x. ln(1) = 0 always; values between 0 and 1 give negative logarithms.
Quick Answer
For the pre-filled example, ln(10) = 2.30258509, with log₁₀(10) = 1.00000000 and log₂(10) = 3.32192809 alongside it. The verification row closes the loop: e^2.302585 = 10.000000. Switch to antilog mode and the same tool runs the film backwards, returning 100.0000000 for a log value of 2 in base 10.
Key takeaways
- One run, four answers: the headline base you picked plus ln, log base 10 and log base 2 in the table, every time.
- The working is shown, not hidden: change-of-base steps under every result, and a verification row that exponentiates the answer back to your input.
- Antilog is a full second mode, not an afterthought, and bases between 0 and 1 work in both directions.
- Your point is plotted on the ln curve, so a negative result for x below 1 reads as a position on the graph, not a surprise.
- Quick Answer
- How to use the natural log calculator, in both modes
- The rules under the hood: change of base does the heavy lifting
- Worked example: ln(10) in three bases, then reversed
- Reading the ln curve, and the three anchor values
- Which base earns its keep where
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the natural log calculator, in both modes
Log mode finds the logarithm of a number; antilog mode recovers the number from a logarithm. Pick the mode first, because it changes which fields the calculator reads.
- Log mode: enter any positive x, choose the headline base (e, 10, 2 or custom), and click Calculate. The headline answer follows your base choice; the table underneath always lists ln, log base 10 and log base 2, so comparisons across bases need no second run. The quick-pick buttons under the x field load e, π, 10, 100 or 0.5 and calculate immediately.
- Custom base: choose Custom base and the base field appears, pre-filled with 3. Any positive base except 1 works, including fractions: log base 0.5 of 10 returns -3.32192809, correctly negative because a base below 1 shrinks as the power grows.
- Antilog mode: enter the log value y and the base it was taken in. The tool returns x = b^y to 10 significant figures, plus the natural log of the result so you can cross-check against the log mode. This is the mode that pairs with the inverse log calculator if you work with antilogs on their own.
- Check the verification row before you trust a number: it raises your base to your result and should hand your x back. If it does not, you have read the wrong row, which is the most common log error there is.
The rules under the hood: change of base does the heavy lifting
Every base in this tool is computed one way: log base b of x equals ln(x) divided by ln(b). That single formula, set out formally in the NIST Digital Library of Mathematical Functions, section 4.2, is why a custom base costs the calculator nothing extra, and it is the same step the results panel prints under your answer.
| Rule | Formula | Checked example |
|---|---|---|
| Definition | ln(x) = y means e^y = x | e^2.30258509 = 10, so ln(10) = 2.30258509 |
| Product rule | ln(a×b) = ln(a) + ln(b) | ln(100) = ln(10) + ln(10) = 4.60517019 |
| Quotient rule | ln(a÷b) = ln(a) - ln(b) | ln(0.5) = ln(1) - ln(2) = -0.69314718 |
| Power rule | ln(x^y) = y × ln(x) | ln(1000) = 3 × ln(10) = 6.90775528 |
| Change of base | log_b(x) = ln(x) ÷ ln(b) | log₂(10) = 2.30258509 ÷ 0.69314718 = 3.32192809 |
Underneath those rules sits one definition worth knowing: ln(x) is the area under the curve 1/t from 1 to x, the integral definition developed in the LibreTexts calculus treatment of the logarithm. That is also why the derivative of ln(x) is exactly 1/x, the fact that makes the natural base unavoidable in calculus while base 10 stays a matter of convenience. Logs undo powers, so if exponent questions are the other half of your homework, the exponent calculator covers that direction.
Worked example: ln(10) in three bases, then reversed
The pre-filled example is x = 10 in base e. Running it gives a headline of 2.30258509, and the step line under the result shows the change of base arithmetic in full: ln(10) ÷ ln(e) = 2.302585 ÷ 1.000000 = 2.30258509.
The table in the same result answers the two follow-up questions students actually ask next: log₁₀(10) = 1.00000000, because 10^1 = 10, and log₂(10) = 3.32192809, because 2^3.32192809 = 10. The log base 2 calculator digs into that binary figure on its own if your work is mostly computing. Four more real runs of this tool, covering the sign change below 1 and a custom base, check out as follows:
| Run | Headline result | ln(x) | log₁₀(x) | log₂(x) |
|---|---|---|---|---|
| x = 1, base e | 0.00000000 | 0.00000000 | 0.00000000 | 0.00000000 |
| x = 0.5, base e | -0.69314718 | -0.69314718 | -0.30103000 | -1.00000000 |
| x = 10, custom base 3 | 2.09590327 | 2.30258509 | 1.00000000 | 3.32192809 |
| x = 1,000,000, base e | 13.81551056 | 13.81551056 | 6.00000000 | 19.93156857 |
Reversing the example is one mode switch: antilog with y = 2.30258509 and base 2.718281828459045 returns 9.999999970, the tiny shortfall coming from the rounded y you typed rather than from the tool. Feed the unrounded chain the other way, log then antilog on the same numbers inside one calculation, and the recovery is exact.
Reading the ln curve, and the three anchor values
After you calculate, the graph under the results plots y = ln(x) with your input as a red dot. Three landmarks make the curve click, and they explain most log behaviour in one picture:
ln(1)
0
Zero at x = 1, for every base
ln(e)
1
The base itself always logs to 1
ln(2)
0.69314718
Each doubling adds exactly this much
ln(10)
2.30258509
The bridge between ln and log base 10
Left of x = 1 the curve dives steeply negative, because shrinking a number toward zero pushes its log toward negative infinity. Right of 1 it climbs but flattens: each doubling of x adds only ln(2) = 0.69314718, never more, which is the whole reason log scales tame enormous ranges. The constant e behind the curve, defined by the limit of (1 + 1/n)^n and documented in the Wolfram Language reference for E, is also the value for which the exponential function is its own derivative.
Which base earns its keep where
Base e is the default in theory, base 10 in measurement, and base 2 in computing. Picking the wrong one is not a style error; it changes the number, so match the base to the field:
- Natural log (e): continuous growth and decay, doubling time and half-life (T = ln(2) ÷ rate), calculus, and most scientific modelling. Growth problems phrased with a rate per year belong here, and the exponential growth calculator picks the thread up from the growth side.
- Common log (10): pH, decibels, the Richter scale and scientific notation, where powers of ten are the natural ruler. Older versions of this page lumped pH and Richter in with natural log uses; they are base 10, and the distinction matters.
- Binary log (2): algorithm runtimes, binary search depth, information in bits, and audio or image doubling questions in computing.
- Custom base: textbook problems and conversions, handled here by change of base rather than by a separate tool per base.
What competing calculators miss
Log tools are plentiful, and most are good at one slice of the job. The recurring gaps, checked against the tools ranking for this keyword:
- Pearson's log calculator: the closest rival, with a base toggle, change-of-base steps and a mini curve. It lives inside a course platform and stops at the logarithm itself: no antilog mode to run the answer backwards and check your equation solving.
- CalculatorX log calculator: broader in one direction, solving for the argument or even the base when a field is left blank. The trade-off is a busier form, and the solved-for-base feature answers a question far fewer people have than "what is ln of this, and what does it reverse to?"
- Engineering log pages (Firgelli and similar): verification is present, which is to their credit, but the natural log beside it is rounded to 2 decimal places (4.61 for ln(100)). At that precision the verification row cannot actually verify anything tightly.
- Single-purpose ln tools (MiniWebTool, Numberbau): clean and fast for ln alone, some with a derivative and a precision selector, but base 10, base 2, custom bases and antilog each need a separate page and a separate trip.
- The gap this page fills: ln, log base 10, log base 2 and any custom base in one run at 8 decimal places, an antilog mode in the same tool, the change-of-base step printed under the result, a verification row, and your point plotted on the curve. No page-hopping between bases, no 2-decimal rounding dressed up as a check.
Frequently asked questions
The natural logarithm, written ln(x), is the logarithm with base e, about 2.71828. It answers one question: to what power must e be raised to get x? So ln(10) = 2.30258509 because e^2.30258509 = 10. It is the inverse of the exponential function e^x, which is why the antilog mode of this calculator reverses it exactly.
ln always means base e. Plain log usually means base 10 on school and engineering calculators, but in much university maths and in several programming languages log means the natural log instead. The notation is genuinely ambiguous, which is why this calculator labels every result with its base rather than relying on convention.
Because no real power of a positive base is zero or negative: e^y is positive for every real y, so no exponent can produce 0 or a negative number. ln(0) and ln of a negative number have no real answer, and the calculator says so with an error instead of inventing one. Complex logarithms exist, but they are outside what this tool computes.
The antilog reverses a logarithm: if y = log base b of x, then x = b^y. Switch the mode selector to Antilog, enter the log value y and the base b, and the calculator returns x. The pre-filled example, y = 2 with base 10, returns 100.0000000, because 10^2 = 100.
By the change of base formula: log base b of x equals ln(x) divided by ln(b). Pick Custom base, enter any positive base other than 1, and the calculator applies that formula. Bases between 0 and 1 are valid and return negative results for x above 1: log base 0.5 of 10 is -3.32192809. Base 1 is rejected, because 1 to any power is 1, so it can never reach x.
ln(1) = 0, because e^0 = 1. ln(e) = 1, because e^1 = e. ln(2) = 0.69314718, the constant behind doubling time and half-life: a quantity growing at rate r doubles in ln(2) divided by r. These three anchor values are worth memorising, since every other ln value falls between landmarks built from them.
It raises the chosen base back to the computed result and shows what comes out, to 8 significant figures. If the verification value matches your x, the logarithm is right. For x = 10 in base e it shows e^2.302585 = 10.000000. It is a cheap self-audit that catches the classic mistake of reading the wrong base row.
Anywhere change is continuous and proportional: compound growth and doubling times, radioactive decay and half-life, cooling, and calculus, where ln(x) is the function whose derivative is 1/x. Not everything logarithmic is natural, though: pH and the Richter scale run on base 10, and algorithm analysis runs on base 2, and this calculator returns both of those alongside ln so the right base is one click away.
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References
- National Institute of Standards and Technology. Digital Library of Mathematical Functions, Chapter 4: Elementary Functions, Section 4.2, Logarithm, Exponential, Powers. Accessed October 22, 2026.
- LibreTexts, Calculus. The Logarithm Defined as an Integral, Section 7.1. Accessed October 22, 2026.
- Wolfram Research. e, from Wolfram MathWorld. Accessed October 22, 2026.
- University of Nebraska-Lincoln, Mathematics. Properties of Logarithms, PreCalculus notes. Accessed October 22, 2026.
- Michigan State University, Mathematics. Natural Logarithms, lecture notes: definition, integral form and properties. Accessed October 22, 2026.
- Wolfram Research. E: Euler's number, base of the natural logarithm, Wolfram Language Documentation. Accessed October 22, 2026.