What is the natural logarithm?

The natural logarithm, written ln(x), is the logarithm with base e (Euler's number ≈ 2.71828). It answers the question: "To what power must e be raised to get x?" For example, ln(e) = 1 and ln(1) = 0.

Why is ln undefined for zero or negative numbers?

The logarithm function is only defined for positive real numbers. There is no real power to which e can be raised to produce zero or a negative number.

What is the difference between ln and log?

ln(x) is base e; log(x) is typically base 10 (common logarithm). You can convert: ln(x) = log(x) / log(e) ≈ log(x) / 0.4343, or equivalently ln(x) = log(x) × 2.302585.

What is the inverse of the natural logarithm?

The inverse of ln(x) is the exponential function e^x. If ln(x) = y, then e^y = x. This calculator shows both values so you can verify your answer.

ln(1) = 0, because e^0 = 1. Any logarithm of 1 (in any base) equals zero.

Can I compute ln of a fraction?

Yes. For a fraction like 0.5, ln(0.5) ≈ -0.6931. Logarithms of numbers between 0 and 1 are always negative.

Natural Log Calculator

Name: Natural Log Calculator
Availability: InStock
Author: Nham Vu

Enter any positive number and instantly get its natural logarithm (ln) with a step-by-step breakdown.

Compute ln(x)

Enter a positive number

Quick examples

Enter a number on the left and click Calculate.

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Summary

Enter any positive number and instantly get its natural logarithm (ln) with a step-by-step breakdown.

How it works

Enter any positive number in the input field.
The calculator evaluates ln(x) using the natural logarithm function.
The result is displayed immediately with full decimal precision.
A step-by-step breakdown shows the key properties and relationships used.
Click "Copy Result" to copy the value to your clipboard.

Use cases

Solving exponential growth and decay problems in science and finance.
Computing entropy in information theory and thermodynamics.
Verifying calculus homework involving ln(x) derivatives and integrals.
Finding the time required for compound interest to reach a target.
Converting between natural log and log base 10 values.
Checking results from statistical models that use log-likelihood.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu