Log Base 2 Calculator

What this calculator does

So, you want the logarithm of a number to base 2, the binary logarithm. This tool takes a positive number and returns its log base 2, worked out to ten decimal places.

There is one input, the number, and the base is fixed at 2. It is the same idea as a general logarithm, locked to the base that matters most in computing.

How to use it

1. Enter your number (x), which must be positive.
2. Press Calculate.

The answer is given to ten decimal places, more precision than most logarithm tools, which is handy when small differences matter.

Log base 2: counting in doublings

The log base 2 of a number is the power of 2 that produces it, which is the same as asking how many times you need to double 1 to reach that number. Start at 1 and double: 1, 2, 4, 8, 16. To get to 8 took three doublings, so the log base 2 of 8 is 3. To reach 16 took four, so log base 2 of 16 is 4. Read the other way, it counts halvings: how many times you can halve a number before you get down to 1. That doubling-and-halving rhythm is exactly what base 2 captures.

Why base 2 lives in computing

Base 2 is the natural language of computers, because a computer stores everything in bits, and each bit doubles how many things you can represent. One bit gives 2 options, two bits give 4, three bits give 8, and so on. So the number of bits you need to label a set of possibilities is the log base 2 of how many there are: 1024 possibilities need 10 bits, because the log base 2 of 1024 is 10. The same logarithm shows up throughout computer science, in the depth of binary trees and in how many steps a binary search takes, which is roughly the log base 2 of the number of items. Whenever something repeatedly splits in half or doubles, base 2 is the log that describes it.

When the answer is not a whole number

Powers of 2 give tidy whole-number logs, but most numbers fall between them, so their logs are decimals. The log base 2 of 10, for instance, is about 3.32, because 10 sits between 8, which is 2 cubed, and 16, which is 2 to the fourth. The decimal simply says how far along you are between those two powers. This is where the tool's ten decimal places earn their keep, giving a precise position rather than a rough one.

A worked example

Enter 8. The tool returns 3, since three doublings of 1 reach 8. Enter 1024 and it returns 10, because 2 to the tenth is 1024. Enter 10 and it returns about 3.3219, placing 10 between the third and fourth powers of 2.

Why the number must be positive

The tool only accepts positive numbers. This is not a quirk of base 2 but a fact about all logarithms: 2 raised to any power, positive, negative, or fractional, always comes out positive, so no power of 2 can ever equal zero or a negative number. Asking for the log of such a value has no answer, which is why the tool asks for a positive input.

Questions people ask

What does log base 2 tell me?

The power of 2 that gives your number, which is how many times you double 1 to reach it, or how many times you can halve the number to get back to 1.

Why is base 2 used in computing?

Because each bit doubles the number of things you can represent, so the bits needed for a set of possibilities is the log base 2 of how many there are.

Why is the answer sometimes a decimal?

Because numbers that are not exact powers of 2 fall between them. The decimal shows how far between two powers the number sits.

Why must the number be positive?

Because no power of 2 is ever zero or negative, so only positive numbers have a base 2 logarithm.

What if I want a different base?

Use the general logarithm calculator for any base, or the base 10 and natural log tools for those specific bases.

References

On the binary logarithm. The base 2 logarithm gives the power of 2 that produces a number, and it is central to computing, where information is measured in bits.

1. Eric W. Weisstein, "Logarithm," from MathWorld, a Wolfram resource, on logarithms including the base 2 logarithm, written lg.
2. "History of logarithms," Wikipedia, on how logarithms turn multiplication into addition, from Napier's tables to modern computing.