# log2

Learn how to use the log2 function in Notion formulas.
The log2() function returns the base-2 logarithm of a number.
1
log2(number)
For reference, here are the named components of a logarithm:
$\log_{base} argument = exponent$
A logarithm is the exponent of the base that returns the argument.

## Example Formulas

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log2(64) // Output: 6
2
3
log2(2) // Output: 1

### Computing Logs with Other Bases (Log Base X)

Notion only provides log2(), log10, and ln because those are by far the most commonly-used log bases.
However, if you need to compute a log with a different base, you can use the change of base formula:
$\log_x y= (\frac{\ln y}{\ln x})$
So, for example:
$\log_3 81 = (\frac{\ln 81}{\ln 3})$
Here’s a Notion formula to compute this:
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ln(81) / ln(3) // Output: 4

## Example Database

$log_{2}$
can be used to perform an interesting mathematical trick – finding the length (i.e. number of digits) in any number when represented in binary. It is done by taking the floor value of a number’s
$log_{2}$
$\lfloor log_{2}n \rfloor + 1$
The example database below shows this trick in action. Unfortunately, actual binary representation of a base-10 number in Notion is not possible (or at least has not been solved by the Notion community), but you can check the result using this converter.

### “Length” Property Formula

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// Compressed
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floor(log2(prop("Num"))) + 1
3
4
// Expanded
5
floor(
6
log2(
7
prop("Num")
8
)
9
) + 1
If you’re curious about the math behind why this trick works, check out this thread:
The formula itself is quite simple:
1. 1.
We pass prop("Num") to the log10() function.
2. 2.
This value is then run through the floor function in order to round it to its nearest integer of lower or equal value.
3. 3.
Finally, we add 1.