/JavaScript

# Math.log1p

The Math.log1p() function returns the natural logarithm (base e) of 1 + a number, that is

$∀x>-1,Math.log1p(x)=ln(1+x)\forall x > -1, \mathtt{\operatorname{Math.log1p}(x)} = \ln(1 + x)$

## Syntax

Math.log1p(x)

### Parameters

x
A number.

### Return value

The natural logarithm (base e) of 1 plus the given number. If the number is less than -1, NaN is returned.

## Description

For very small values of x, adding 1 can reduce or eliminate precision. The double floats used in JS give you about 15 digits of precision. 1 + 1e-15 = 1.000000000000001, but 1 + 1e-16 = 1.000000000000000 and therefore exactly 1.0 in that arithmetic, because digits past 15 are rounded off.

When you calculate log(1 + x), you should get an answer very close to x, if x is small (that's why these are called 'natural' logarithms). If you calculate Math.log(1 + 1.1111111111e-15) you should get an answer close to 1.1111111111e-15. Instead, you will end up taking the logarithm of 1.00000000000000111022 (the roundoff is in binary so sometimes it gets ugly), so you get the answer 1.11022...e-15, with only 3 correct digits. If, instead, you calculate Math.log1p(1.1111111111e-15) you will get a much more accurate answer 1.1111111110999995e-15 with 15 correct digits of precision (actually 16 in this case).

If the value of x is less than -1, the return value is always NaN.

Because log1p() is a static method of Math, you always use it as Math.log1p(), rather than as a method of a Math object you created (Math is not a constructor).

## Polyfill

This can be emulated with the following function:

Math.log1p = Math.log1p || function(x) {
x = Number(x);
if (x < -1 || x !== x)
return NaN;
if (x === 0 || x === Infinity)
return x;

var nearX = (x + 1) - 1;

return nearX === 0 ? x : x * (Math.log(x + 1) / nearX);
};


## Examples

### Using Math.log1p()

Math.log1p(1);  // 0.6931471805599453
Math.log1p(0);  // 0
Math.log1p(-1); // -Infinity
Math.log1p(-2); // NaN


## Browser compatibilityUpdate compatibility data on GitHub

Desktop
Chrome Edge Firefox Internet Explorer Opera Safari
log1p 38 12 25 No 25 8
Mobile
Android webview Chrome for Android Firefox for Android Opera for Android Safari on iOS Samsung Internet
log1p 38 38 25 25 8 3.0
Server
Node.js
log1p 0.12