Using np.frexp(arr)[1] comes in 4 to 6x faster than np.ceil(np.log2(x)).astype(int).

Note that, as pointed out by @GregoryMorse above, some additional work is needed to guarantee correct results for 64-bit inputs (bit_length3 below).

import numpy as np


def bit_length1(arr):
    # assert arr.max() < (1<<53)
    return np.ceil(np.log2(arr)).astype(int)

    
def bit_length2(arr):
    # assert arr.max() < (1<<53)
    return np.frexp(arr)[1]


def bit_length3(arr):  # 64-bit safe
    _, high_exp = np.frexp(arr >> 32)
    _, low_exp = np.frexp(arr & 0xFFFFFFFF)
    return np.where(high_exp, high_exp + 32, low_exp)

Performance results, 10 iterations on a 100,000-element array via https://perfpy.com/868.