Natural Logarithm Calculator

The natural logarithm answers one specific question: what power do you raise $e$ to in order to get a given number? Here $e$ is Euler's number, roughly 2.71828, and the answer is written $\ln(x)$. This calculator returns $\ln(x)$ for any positive number you enter, and it works the other direction too, give it a logarithm and you get back the original value through $e^x$.

What is the natural logarithm?

$\ln(x)$ is the logarithm with base $e$. The "natural" part of the name comes from how the function drops out of calculus without anyone forcing it there. The derivative of $\ln(x)$ is $1/x$, and the integral of $1/x$ is $\ln(x)$. Pick any other base and a constant tags along; $e$ is the base that keeps both sides clean.

A few values are worth keeping in your head. $\ln(1) = 0$ because $e^0 = 1$. $\ln(e) = 1$ because $e^1 = e$. $\ln(e^2) = 2$. Anything between 0 and 1 gives a negative result ($\ln(0.5) \approx -0.693$), and zero or negative inputs are undefined on the real line.

How to use this calculator

Put a positive number into $x$ to read off $\ln(x)$, or put a known logarithm into the $\ln(x)$ field to recover the original number. For instance, $x = 20$ gives $\ln(20) \approx 2.996$. Type 2.996 into $\ln(x)$ and 20 comes back, since $e^{2.996} \approx 20$. So the same tool handles direct evaluation and exponential equations like $e^t = 20$ without switching modes.

Where natural logarithms show up

Natural logs turn up across math, science, and engineering. In calculus they appear the moment a rational function has $x$ in the denominator, every integral of $1/x$ lands on $\ln(x)$. For population growth, radioactive decay, and continuously compounded interest, $\ln(2)$ gives doubling time or half-life directly through $t = \ln(2)/r$. Statisticians use them for log-normal distributions and for taming the variance of skewed data, and in physics the time constants of RC circuits, the entropy term in thermodynamics, and signal attenuation in decibels all run on natural logs.

Identities worth remembering

A small handful of identities make natural logs much easier to work with.

• Multiplication becomes addition: $\ln(ab) = \ln(a) + \ln(b)$. This is the trick that made slide rules useful for a hundred years.

• Powers come out front: $\ln(x^n) = n \cdot \ln(x)$. Handy whenever an exponent is the unknown.

• A negative answer means the input sits between 0 and 1. $\ln(0.5)$ is about -0.693, and $\ln(0.1)$ is about -2.303.

• Reach for $\ln$ when you're doing calculus or modeling continuous growth. Reach for $\log_{10}$ when you're talking about orders of magnitude or scientific notation.

Frequently asked questions

What is the difference between ln and log?

$\ln$ is the natural logarithm, base $e \approx 2.718$. "log" on its own usually means base 10, but the convention shifts by field: computer scientists often mean base 2, and plenty of calculus textbooks use log to mean $\ln$. When the base matters, check the source's convention.

Why is it called the 'natural' logarithm?

Because it shows up in calculus on its own. The derivative of $\ln(x)$ is just $1/x$, and the integral of $1/x$ is just $\ln(x)$. Choose any other base and an extra constant clings to every result. The base $e$ is the one that lets you write the calculus without bookkeeping.

Can I take the natural log of a negative number?

Not on the real numbers. $\ln(x)$ is defined only for $x > 0$, and $\ln(0)$ runs off to negative infinity. Negative inputs do have a logarithm in the complex numbers ($\ln(-1) = i\pi$, for instance), but most calculators don't return one, and this one doesn't either.

How precise is the answer?

The math runs in double-precision floating point, so the result is accurate to about 15 or 16 significant digits. That's well past anything you'd need outside specialized numerical work.