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floorlogb

floorlogb is the function that returns the greatest integer n such that b^n ≤ x for a positive real x and a base b > 1. Equivalently, floorlogb(x) = ⌊log_b(x)⌋, where log_b is the logarithm with base b. The function is defined for x > 0 and b > 1; log bases b ≤ 0 or b = 1 are not allowed in the standard definition.

For integer values, floorlogb(n) gives the position of the most significant digit of n in base b,

In the special case b = 2, floorlog2(n) equals the index of the most significant 1 bit in

Computationally, floorlogb(x) can be obtained by evaluating the real logarithm and applying the floor function, ⌊log_b(x)⌋,

See also: logarithm, digit count, bit length.

and
is
related
to
the
number
of
digits.
Specifically,
for
integer
n
≥
1,
n
lies
in
the
interval
[b^k,
b^{k+1})
if
and
only
if
floorlogb(n)
=
k.
Consequently,
the
number
of
base-b
digits
of
n
is
floorlogb(n)
+
1.
the
binary
representation
of
n
(counting
from
0).
For
example,
floorlog2(1)
=
0,
floorlog2(2)
=
1,
floorlog2(7)
=
2,
and
floorlog10(999)
=
2.
or
by
repeated
division
of
x
by
b
in
integer
arithmetic
until
the
quotient
is
less
than
b,
counting
the
steps.