Logarithm calculator

What it is, without formulas

A logarithm answers a single question: how many times do I have to multiply the base by itself to reach this number? The logarithm to base 10 of 1000 is 3, because 10×10×10 makes 1000. The logarithm to base 2 of 1024 is 10, because doubling ten times from 1 gets you to 1024. It's the inverse of raising to a power, exactly as division is the inverse of multiplication.

The three bases you meet

Base 10 is the one behind decibels, pH and the Richter scale: they're all scales where going up by one means multiplying by ten, which is why a magnitude 6 earthquake isn't «a bit stronger» than a magnitude 5 one.

Base 2 is the one computing uses: it says how many bits you need to tell a given number of cases apart (with 10 bits you count 1024 values) and how many times something can be halved. Base e ≈ 2.718 is the natural logarithm, the one that turns up by itself in every growth and decay formula; to use it here just write 2.718 in the base field.

What it's for in real life

For the question «how many years until it doubles?». A sum earning 5% a year doubles in log(2) ÷ log(1.05) years, a little over fourteen. The same arithmetic works for anything growing by a fixed percentage: a subscription going up, a debt with interest, the number of subscribers to a channel.

When the logarithm doesn't exist

It doesn't exist for zero and negative numbers: no exponent can take a positive base to a negative number or to zero. And there's no base 1: one multiplied by itself is always one, so you get nowhere. In all three cases no result appears here, rather than a meaningless number.

Nearby tools

The inverse operation is Power calculator. For complete expressions there is Scientific calculator, and to see how much a sum really grows over the years Compound interest.