# logarithm – Log to the base 2 in python

## logarithm – Log to the base 2 in python

Its good to know that

but also know that

`math.log`

takes an optional second argument which allows you to specify the base:

```
In [22]: import math
In [23]: math.log?
Type: builtin_function_or_method
Base Class: <type builtin_function_or_method>
String Form: <built-in function log>
Namespace: Interactive
Docstring:
log(x[, base]) -> the logarithm of x to the given base.
If the base not specified, returns the natural logarithm (base e) of x.
In [25]: math.log(8,2)
Out[25]: 3.0
```

Depends on whether the input or output is `int`

or `float`

.

```
assert 5.392317422778761 == math.log2(42.0)
assert 5.392317422778761 == math.log(42.0, 2.0)
assert 5 == math.frexp(42.0)[1] - 1
assert 5 == (42).bit_length() - 1
```

# float → float `math.log2(x)`

```
import math
log2 = math.log(x, 2.0)
log2 = math.log2(x) # python 3.3 or later
```

- Thanks @akashchandrakar and @unutbu.

# float → int `math.frexp(x)`

If all you need is the integer part of log base 2 of a floating point number, extracting the exponent is pretty efficient:

```
log2int_slow = int(math.floor(math.log(x, 2.0))) # these give the
log2int_fast = math.frexp(x)[1] - 1 # same result
```

- Python frexp() calls the C function frexp() which just grabs and tweaks the exponent.
- Python frexp() returns a tuple (mantissa, exponent). So
`[1]`

gets the exponent part. - For integral powers of 2 the exponent is one more than you might expect. For example 32 is stored as 0.5×2⁶. This explains the
`- 1`

above. Also works for 1/32 which is stored as 0.5×2⁻⁴. - Floors toward negative infinity, so log₂31 computed this way is 4 not 5. log₂(1/17) is -5 not -4.

# int → int `x.bit_length()`

If both input and output are integers, this native integer method could be very efficient:

```
log2int_faster = x.bit_length() - 1
```

`- 1`

because 2ⁿ requires n+1 bits. Works for very large integers, e.g.`2**10000`

.- Floors toward negative infinity, so log₂31 computed this way is 4 not 5.

#### logarithm – Log to the base 2 in python

If you are on python 3.3 or above then it already has a built-in function for computing log2(x)

```
import math
finds log base2 of x
answer = math.log2(x)
```

If you are on older version of python then you can do like this

```
import math
finds log base2 of x
answer = math.log(x)/math.log(2)
```