The natural logarithm, written ln, is a logarithm with base e, and e is about 2.71828. ln(x) answers the question “e raised to what power gives x?”, so ln(100) ≈ 4.60517018599 because e^4.60517018599 is 100. It is called natural because e shows up on its own whenever something grows or decays in proportion to its own size.

Key Takeaways

ln is log_e: it asks “e to what power gives x?”, with e ≈ 2.71828.

ln(1) = 0 and ln(e) = 1; ln(100) = 4.60517018599.

e^2 = 7.38905609893, and ln of that value returns exactly 2.

Doubling time at a continuous rate r is ln(2) ÷ r ≈ 0.693 ÷ r.

What ln means

Every logarithm hides a power question. log(100) = 2 because 10² = 100. The natural logarithm asks the same question with e as the base: ln(100) = 4.60517018599 because e^4.60517018599 = 100. Nothing about the idea changes, only the base. The what is a logarithm post covers the general idea, and the natural log calculator does the arithmetic for you.

The value of e, 2.718281828459045, is irrational, so its digits never end and never repeat. You never type the whole thing. The calculator has an e key, and you write powers of it as e^2, e^x, or use the exp button.

The two anchor points: ln(1) = 0 and ln(e) = 1

Two values come up so often they are worth memorising. Any base to the power 0 gives 1, so the log of 1 is 0 in every base, including e: ln(1) = 0. And e to the power 1 is e itself, so ln(e) = 1. Type ln(1) and ln(e) on the calculator and you get exactly 0 and 1.

Worked example 1: ln(100). On the natural log calculator or the main calculator, type ln(100). The calculator returns 4.60517018599. You can cross-check with the exponent key: e^4.60517018599 comes back close to 100, because a logarithm is the inverse of the exponential.

Why the base is e

e looks like a random number until you watch it appear in one famous limit. Start with one unit growing at 100% per year. Compounded once, you end up with 2. Compound twice a year: (1 + 1/2)² = 2.25. Monthly: (1 + 1/12)¹² = 2.61304. Daily: (1 + 1/365)³⁶⁵ = 2.71457. Push the compounding toward its limit, and the answer is exactly e ≈ 2.71828.

That limit matters because it marks the distinctive property of e: the function e^x is its own derivative, so its rate of change at every point equals its value there. No other base has that property, and any smooth exponential growth can be rewritten in terms of e. That makes ln a general measuring stick for growth. The log calculator handles the base-10 version and lets you pick any other base when you need it.

Worked example 2: e² and its log. Type e^2. The calculator returns 7.38905609893. Now use the ln key on the same value, or type ln(7.38905609893). The result drops back to 2. Undo and redo that round trip; it is the practical definition of a logarithm.

Logs of fractions come out negative

ln(1) = 0, ln(e) = 1, ln(e²) = 2, and the values keep climbing. What happens below 1? Numbers between 0 and 1 are powers of e with negative exponents, and their logs are negative. ln(0.5) = −0.693147, because e^(−0.693147) = 0.5. A negative log is not an error. It just means the input is a fraction less than 1.

Decay formulas rely on this. When a sample decays to half its size, the multiplier is 0.5, and ln(0.5) = −0.693147 is what you would take logs of in the exponent.

Worked example 3: fraction inputs. Type ln(0.5). The result is −0.69314718056. Now type ln(2), which returns 0.69314718056. The two are negatives of each other, because 0.5 = 2⁻¹ and the power rule pulls the −1 out in front of the log.

Natural logs and growth and decay

Anything that grows in proportion to its current size follows A = A₀ × e^(rt), where A₀ is the starting amount, r is the rate, and t is time. Take the log of both sides and you get rt = ln(A ÷ A₀), which is why ln answers growth questions directly. Doubling time is the cleanest example. Doubling means e^(rt) = 2, so rt = ln(2), and ln(2) ≈ 0.693.

Worked example 4: doubling time. You want to know how long a population takes to double at a continuous 7% growth rate. Type ln(2) / 0.07. The calculator returns 9.90210257943, so about 9.9 years. This is the source of the rule of 70: divide 70 by the percentage rate to get the doubling time in years, because 0.693 expressed as a percentage sits close to 70.

ln versus log on this calculator

This site follows the scientific calculator convention: log is base 10 and ln is base e. Many programming languages and papers use log for the natural log, so check the convention before you copy an answer. The two are related by a constant factor: ln(x) = log(x) × 2.302585. If you have a base-10 log and you need the natural version, that single multiplication converts it, and the exponent calculator can verify the round trip.

Using natural logs on the calculator

Press the ln key and the calculator inserts ln( automatically. Type the number and close the bracket.
To raise e to a power, use the e key with ^, for example e^3.
Keep the whole expression on one line, parenthesise when you are not sure, and the order of operations sorts out the rest.

Frequently asked questions

Why is ln(1) equal to 0?

Because any nonzero base raised to the power 0 equals 1. A logarithm asks “what exponent produces this number?”, so the log of 1 is 0 in every base, not just e.

What is the difference between ln and log?

The base. ln uses e ≈ 2.71828, and log on a scientific calculator uses 10. They are proportional: ln(x) ≈ 2.302585 × log(x). This calculator follows the physics convention where log means base 10.

Can the natural log take a negative number?

No, not with real numbers. eⁿ is positive for every real n, so there is no real power of e that produces a negative input, and the calculator reports an undefined result.

How do I type e on the calculator?

Use the e key for the constant and the ^ key for powers: e^2 gives 7.38905609893. For the exponential button, the exp key takes a number and returns e raised to it.