Natural Log Calculator

What this calculator does

So, you want the natural logarithm of a number, its log to base e. This tool takes a positive number and returns its natural log, written ln, to five decimal places.

There is one input, the number, with the base fixed at e, the special constant roughly equal to 2.71828. It is the logarithm that mathematics and science reach for by default.

How to use it

1. Enter your number (x), which must be positive.
2. Press Calculate.

The natural log: base e

The natural logarithm of a number is the power of e that produces it. So the natural log of e itself is 1, since e to the power 1 is e, and the natural log of 1 is 0, since e to the power 0 is 1. The number e is a fixed constant, about 2.71828, that turns up throughout mathematics whether we invite it or not. Written ln rather than log, the natural logarithm is the log built around this particular base.

Why base e is the natural one

Base 10 was chosen to match our fingers; base e is chosen by the mathematics itself. Its defining feature shows up in calculus: the natural logarithm is the one whose rate of change is exactly 1 over x, the simplest possible form. Every other base drags an untidy constant into that rate, but base e keeps it clean. This is why, once you move past routine arithmetic into anything involving rates, areas, or continuous change, base e appears on its own accord. It is less a choice than a discovery about how growth and change actually work.

Growth, decay, and the natural log

The natural log is the tool for anything that grows or shrinks continuously, at a rate proportional to its current size. Money under continuous compound interest, a population multiplying, a radioactive sample decaying, a hot drink cooling: all of these follow curves built on e, and the natural log is what unwinds them. If you know how much something has grown and want the time or the rate behind it, the natural log is the step that gets you there. That is why it sits at the heart of the formulas for continuous growth, decay, and half-life.

A worked example

Enter e, roughly 2.71828. The tool returns 1, because e to the power 1 is e. Enter 1 and it returns 0. Enter about 7.389 and it returns close to 2, because e squared is about 7.389. As with any logarithm, numbers below 1 give negative results and numbers above 1 give positive ones.

Why the number must be positive

The tool accepts only positive numbers, for the same reason every logarithm does. The constant e raised to any power stays positive, so no power of e can reach zero or go negative. A natural log of zero or of a negative number therefore has no value, and the tool asks for a positive input to keep the answer meaningful.

Questions people ask

What does the natural log tell me?

The power of e that gives your number. The natural log of e is 1, and of 1 is 0.

What is e?

A fixed mathematical constant, about 2.71828, that arises naturally in growth, calculus, and many other areas.

Why is base e called natural?

Because its rate of change is exactly 1 over x, the simplest of any base. Other bases carry an extra constant, so e is the one the mathematics singles out.

Where does the natural log come up?

In anything growing or decaying continuously: compound interest, populations, radioactive decay, cooling. It is the step that unwinds those e-based curves.

Why must the number be positive?

Because no power of e is ever zero or negative, so only positive numbers have a natural logarithm.

References

On the natural logarithm. The natural log has base e, the constant about 2.71828, and it is favoured in calculus because its rate of change is simply 1 over x.

1. Eric W. Weisstein, "Natural Logarithm," from MathWorld, a Wolfram resource, on the logarithm to base e and its simple derivative.
2. Eric W. Weisstein, "Ln," from MathWorld, a Wolfram resource, on the ln notation for the logarithm to base e.