Why is there no logarithm of zero?

Because no power of a positive base ever reaches zero. It only approaches it, so the logarithm runs to negative infinity rather than landing on a value.

When should I use natural log rather than base ten?

Whenever something grows or decays continuously. The natural base is what makes the rate of change equal to the quantity itself, which is why it turns up in compound interest and half-lives.

Math & Statisticsalgebra

Logarithm Calculator

A logarithm in any base, computed through the change-of-base identity. A logarithm answers "what power do I raise this base to", which is the sentence that makes the whole idea click.

Also called: log calculator, ln calculator, natural log.

Logarithm

log base 10 of 1,000 is 3, which means 10 raised to 3 gives 1,000. In natural logs that is 6.91, and in base two it is 9.97.

Natural log

6.91

Log base 10

Log base 2

9.97

Check

10 ^ 3 = 1000

How this is calculated

Method and background

How this is calculated

Every logarithm is the same question in different clothing: to what power must the base be raised to get this number? Because computers only carry natural logs, any base is found by dividing one natural log by another, which is the change-of-base rule. The three bases in common use are ten for orders of magnitude, e for anything growing continuously, and two for anything that doubles.

log base b of x = natural log of x / natural log of b

x: The number
b: The base

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

log base ten of a thousand

Number: 1,000
Base: 10

Logarithm3

arithmetic identity

Open this example

the natural log of e is one

Number: 2.72
Base: 2.72

Logarithm1

boundary: the definitional case

Open this example

Method and limits

What it assumes

A positive number and a positive base other than one.

What it deliberately does not model

The logarithm of zero or a negative number is undefined in the reals and is declined.
Base one has no logarithm, because one raised to any power is still one.

Formula version 1.0.0 · definition 1.0.0 · United States · Report a problem with this calculator

Frequently asked questions

Why is there no logarithm of zero?: Because no power of a positive base ever reaches zero. It only approaches it, so the logarithm runs to negative infinity rather than landing on a value.
When should I use natural log rather than base ten?: Whenever something grows or decays continuously. The natural base is what makes the rate of change equal to the quantity itself, which is why it turns up in compound interest and half-lives.