The Natural Log Calculator (ln)
The natural log calculator (or simply ln calculator) determines the logarithm to the base of a famous mathematical constant, e, an irrational number with an approximate value of e = 2.71828. In other words, it calculates the natural logarithm.
But what is the natural logarithm, ln x, of a given number x? This is the power the number e has to be raised to in order to result in a given number x.
How to use the natural logarithm calculator
Like all other logarithms, the natural logarithm of x returns the power, or exponent, to which a given base e must be raised to yield back the number x. It is easier to understand this notion when the base is an integer, for example, 2 or 3:
- log2 16 = 4 since 24 = 16
- log3 81 = 4 since 34 = 81
In the case of the natural logarithm, this is somewhat less intuitive because its base, e, is not an integer. But since the value of e is between 2 and 3, we understand that e4 has to be somewhere between 24 = 16 and 34 = 81. It turns out that e4 = 54.498, which we can write as:
Here are some examples of the natural logarithm — check them with the calculator above:
| Expression | Value | Because |
|---|---|---|
| ln 1 | 0 | e0 = 1 |
| ln 10 | 2.3026 | e2.3026 = 10 |
| ln 20 | 2.9957 | e2.9957 = 20 |
| ln 50 | 3.9120 | e3.9120 = 50 |
| ln 100 | 4.6052 | e4.6052 = 100 |
Other ways to denote the natural logarithm
One way of denoting the natural logarithm is loge. This is the same as when we write the logarithm to the base two as log2. A more common way to write it is ln, an abbreviation of the Latin expression logarithmus naturalis — the name the natural logarithm was given when Latin was still the lingua franca of science.
There is also a third notation: plain log. This is somewhat problematic, as it is often mistaken for the logarithm to the base 10. However, this is the syntax many software implementations of the natural logarithm use, so be careful!
What's so natural about the natural logarithm?
Maybe the most important property of the natural logarithm is that it is the inverse function of the exponential function ex — the only function whose rate of change, or derivative, is precisely itself: (ex)' = ex. In simpler words, the exponential function governs its own rate of change, which makes both ex and its inverse ln x the natural choice when describing many real-world phenomena.
The natural log graph
Another interesting property is how ln x changes as the argument x increases. The derivative of the natural logarithm is inversely proportional to x:
Although ln x increases as x increases, the rate of growth gets smaller and smaller as x approaches higher and higher values. The curve passes through (1, 0), heads to −∞ as x approaches 0 from the right, and grows without bound — but ever more slowly — as x → ∞.
Where did the number e come from?
The constant e ≈ 2.718281828 is the base of the natural logarithm. It arises naturally from the limit that describes continuous growth:
How to make e amount of money
Imagine you invest a principal P at an annual interest rate r, compounded continuously. After time t your balance is:
To find how long it takes to grow your money by a factor x (that is, until A / P = x), just take the natural log of both sides:
The financial application panel in the calculator uses exactly this formula, so you can enter a principal and a continuous rate (in your chosen currency — USD, RUB, EUR and more) and see how many years the growth would take.
The real-world importance of ln 2 and other natural logarithms
Setting the growth factor to x = 2 gives the famous doubling time: t = ln(2) / r ≈ 0.6931 / r. This is the exact continuous-compounding version of the well-known "rule of 70/72" used to estimate how quickly an investment (or a population) doubles.
- ln 2 ≈ 0.6931 — doubling, half-life of radioactive decay
- ln 3 ≈ 1.0986 — tripling time
- ln 10 ≈ 2.3026 — the factor between natural and common logs (log10 x = ln x / ln 10)
Other applications of the natural logarithm
- Calculus — the integral ∫ dx/x = ln|x| + C
- Physics — radioactive decay, RC-circuit discharge, Newton's law of cooling
- Chemistry — the Arrhenius equation and reaction kinetics
- Statistics — log-likelihood, entropy, and the log-normal distribution
- Computer science — natural log appears in complexity analysis alongside log2
Frequently Asked Questions
What is ln of 1?
ln(1) = 0, because e0 = 1. The natural log of 1 is zero for the same reason any logarithm of 1 is zero.
What is ln of e?
ln(e) = 1, because e1 = e. The natural log and the exponential function are inverses of each other.
Can you take the natural log of a negative number or zero?
No. The natural logarithm is only defined for positive numbers (x > 0). As x approaches 0 from the right, ln(x) tends to −∞, and ln(0) and ln of a negative number are undefined in the real numbers.
How do I convert ln to log base 10?
Use the change-of-base formula: log10(x) = ln(x) / ln(10), where ln(10) ≈ 2.302585. The calculator shows both log10 and log2 next to your ln result.
What is the difference between ln and log?
ln is the logarithm to base e (≈ 2.71828), while log usually means the logarithm to base 10. They are proportional: ln(x) = log(x) × ln(10).
References
- Weisstein, Eric W. "Natural Logarithm." MathWorld — A Wolfram Web Resource.
- Abramowitz, M. & Stegun, I. A. Handbook of Mathematical Functions, Chapter 4: Elementary Transcendental Functions.