Math
Natural Log Calculator
Compute ln(x), the logarithm with base e ≈ 2.71828, or switch to e^x mode to undo it.
Powers of e are detected so ln(e²) shows exactly 2, and every answer comes with the equivalent log₁₀ and log₂ values, a check by exponentiation, and the series that defines e^x.
For ln(x), x must be positive. For e^x, any real x works.
Try an example
Result
Result
—
- Expression
- —
- log₁₀ equivalent
- —
- log₂ equivalent
- —
- Check
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the natural logarithm is
The natural The exponent you would need to raise a base to in order to get a number. ln(x) is the logarithm whose The number being raised to a power, or the side a shape's height is measured from. is Euler's number e ≈ 2.718281828. It answers: e to what power gives x? It is called natural because e^x is the one exponential whose rate of change equals its own value, which makes ln the logarithm that appears in calculus, continuous growth, and physics.
Useful natural log rules
| Rule | Formula | Example |
|---|---|---|
| Product | ln(ab) = ln a + ln b | ln 6 = ln 2 + ln 3 |
| Quotient | ln(a/b) = ln a - ln b | ln 0.5 = ln 1 - ln 2 = -0.693 |
| Power | ln(a^k) = k ln a | ln 8 = 3 ln 2 |
| Change of base | log_b(x) = ln x / ln b | log₁₀ 20 = ln 20 / ln 10 |
The series behind e^x
e^x equals the infinite sum 1 + x + x²/2! + x³/3! + ... . With x = 1 the sum gives e itself. The e^x mode shows the first six terms so you can see how quickly the partial sums approach the true value for small x.
How to use it
- Choose ln(x) to take a natural log, or e^x to compute an exponential.
- Enter x. For ln, x must be positive.
- Read the result, then the log₁₀ and log₂ equivalents and the exponentiation check.
- Open the work to see the series terms or the power-of-e detection.
How to read the answer
ln(x) tells you what power of e gives x. ln(1) = 0, ln(e) = 1, and ln grows slowly: ln(1,000,000) is only about 13.8. e^x does the reverse and grows extremely fast.
Common mistakes and edge cases
- ln(0) is undefined and ln of a negative number is not a real number.
- ln(x) is not log₁₀(x). Divide ln by 2.302585 to convert.
- ln(a + b) is not ln a + ln b. The product rule applies to multiplication only.
- Typing e as 2.718 instead of 2.718281828 makes ln(e) come out as 0.99990 instead of 1.
Worked examples
ln of 20
A number that is not a power of e
Result
2.99573227
ln of e
Exact: e^1 = e
Result
1
ln of 1
Any log of 1 is 0
Result
0
ln of a fraction
ln(0.5) is negative
Result
-0.69314718
e^2
Exponential mode
Result
7.3890561
e^-1
Negative exponent gives 1/e
Result
0.36787944
Undefined
ln of a negative number
Result
Error
Frequently asked questions
What is ln?+
ln is the natural logarithm, the logarithm with base e ≈ 2.71828. ln(x) is the exponent you put on e to get x.
What is the difference between ln and log?+
ln uses base e; log on most calculators and in science uses base 10. They are proportional: ln(x) = 2.302585 × log(x). In many math and programming contexts, log with no base means ln.
What is ln(e)?+
1, because e^1 = e. Likewise ln(e²) = 2 and ln(1/e) = -1.
Why is ln(0) undefined?+
No power of e equals 0. As x shrinks toward 0, ln(x) heads to negative infinity, so there is no value to assign.
How do I solve e^x = 50?+
Take ln of both sides: x = ln(50) ≈ 3.912. Use ln mode with x = 50.
How do I convert ln to log base 10?+
Divide by ln(10) ≈ 2.302585: log₁₀(x) = ln(x) / 2.302585. The calculator shows this conversion on every ln result.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
Related calculators
e Calculator
Get Euler's number e to any number of decimal places, compute e^x or the x-th root of e, and see the limit definition (1 + 1/n)^n and the series 1 + 1/1! + 1/2! + ... converge.
Logarithm Calculator
Calculate logarithms in any valid base with common log, natural log, and change-of-base steps.
Log Base 2 Calculator
Calculate log base 2 of any positive number with an exact answer for powers of 2, the decimal value otherwise, and the bits and doublings interpretation.
Change of Base Formula Calculator
Evaluate log_b(x) in any base with the change of base formula, showing both the ln and log₁₀ routes, exact answers when x is a power of b, and every step.
Exponential Growth Calculator
Solve exponential growth for the final amount, starting amount, rate, or time using y = a(1 + r)^t or y = a·e^(kt), with doubling time, a table of values, and a graph.
Adding Fractions Calculator
Add up to five fractions or mixed numbers. Finds the least common denominator with prime factors, rewrites each fraction, adds, simplifies, and shows the mixed-number and decimal forms with a fraction-strip picture.
Last updated: September 4, 2026