The Scientific Calculator

Natural Log Calculator

The natural logarithm uses the constant e ≈ 2.71828 as its base. That looks arbitrary until you see where it comes from.

Last updated: August 2026 · Written and verified by Akash Pandey

Calculate ln or eˣ

Natural logarithm

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This runs the same calculation engine as the main scientific calculator, entirely in your browser. Nothing you type is sent anywhere.

What makes e "natural"

e is not a number someone chose for convenience. It falls out of a concrete question: if you invest £1 at 100% annual interest, how much do you have after a year if the interest is compounded more and more often?

CompoundedCalculationResult
Annually(1 + 1/1)¹2
Quarterly(1 + 1/4)⁴2.44141
Monthly(1 + 1/12)¹²2.61304
Daily(1 + 1/365)³⁶⁵2.71457
Hourly(1 + 1/8760)⁸⁷⁶⁰2.71813
Continuouslythe limite ≈ 2.71828
The sequence converges rather than growing without bound - and its limit is e.

That limit is the definition of e, and it explains why e appears in every setting where something grows or decays at a rate proportional to its current size: compound interest, population growth, radioactive decay, capacitor discharge, and cooling.

ln(x) = n ⟺ eⁿ = x, where e ≈ 2.718281828459

  • ln is simply log_e - a logarithm like any other, with e as its base.
  • The defining property: the function eˣ is its own derivative. No other base does this.

Worked examples

  1. Evaluate ln(e)

    ln(e)

    1. The question is: e to what power gives e?
    2. Any number to the power 1 is itself.
    3. This is the general rule log_b(b) = 1 applied to base e.

    = 1

  2. Evaluate ln(100)

    ln(100)

    1. e to what power gives 100?
    2. e² ≈ 7.389 and e⁵ ≈ 148.4, so the answer sits between 4 and 5.
    3. The calculator gives approximately 4.60517.
    4. Cross-check with the log laws: ln(100) = 2 × ln(10) ≈ 2 × 2.302585. ✓

    = 4.60517018599

  3. How long does it take to double at 7% continuous growth?

    ln(2) / 0.07

    1. Continuous growth follows A = A₀ · e^(rt). Doubling means e^(rt) = 2.
    2. Take ln of both sides: rt = ln(2) ≈ 0.693147.
    3. Solve for t: t = 0.693147 ÷ 0.07.
    4. This is where the "rule of 70" comes from - ln(2) is about 0.7, so 70 divided by the percentage rate gives the doubling time.

    = 9.90210257943

  4. Evaluate e^1

    exp(1)

    1. This simply returns the constant e itself.
    2. The calculator shows it to twelve significant digits.

    = 2.71828182846

ln versus log

The two differ only by a constant factor, because of the change of base rule: ln(x) = log(x) ÷ log(e) ≈ log(x) × 2.302585. Any logarithm can be converted to any other by a single multiplication, which is why the choice of base rarely changes the shape of a result - only its scale.

xln(x)log₁₀(x)
0.5−0.693147−0.301030
100
20.6931470.301030
e ≈ 2.7182810.434294
31.0986120.477121
51.6094380.698970
102.3025851
503.9120231.698970
1004.6051702
Values rounded to six decimal places.

Frequently Asked Questions

What is the difference between ln and log?

They use different bases. ln is base e ≈ 2.71828; log on a scientific calculator is base 10. They are proportional: ln(x) ≈ 2.302585 × log(x).

What is the value of e?

e ≈ 2.718281828459045. It is irrational and transcendental, so its decimal expansion never repeats or terminates. It is defined as the limit of (1 + 1/n)ⁿ as n grows without bound.

Why is e called the natural base?

Because eˣ is the only exponential function that is its own derivative - its rate of change at every point equals its value there. That makes it the natural description of anything growing in proportion to its own size.

What is ln(1)?

Zero. Any base raised to the power 0 gives 1, so the logarithm of 1 is 0 in every base.

Can ln take a negative number?

Not within the real numbers. eⁿ is positive for every real n, so no real logarithm produces a negative argument. The calculator reports an undefined result.

What is the rule of 70?

A shortcut for doubling time: divide 70 by the percentage growth rate. It works because doubling requires ln(2) ≈ 0.693, and 0.693 as a percentage is close to 70.

How does ln relate to continuous compounding?

If you invest at rate r compounded continuously, the growth factor after t years is eʳᵗ. Taking the natural log gives rt, which is the continuously compounded return. This is why ln appears throughout finance.

What is ln(e)?

Exactly 1. Since ln asks "what power of e gives this number?", and e¹ = e, the natural logarithm of e is 1 by definition.