Python math.log详解：对数运算log2/log10/log1p应用

import math

math.log(x[, base])    # 自然对数或指定底的对数
math.log2(x)           # 以2为底的对数
math.log10(x)          # 以10为底的对数
math.log1p(x)          # log(1+x)，x接近0时更精确

import math

# 自然对数
print(f"ln(e)  = {math.log(math.e)}")
print(f"ln(1)  = {math.log(1)}")
print(f"ln(10) = {math.log(10):.6f}")

# 指定底数
print(f"\nlog2(8)   = {math.log(8, 2)}")
print(f"log10(100) = {math.log(100, 10)}")
print(f"log3(81)  = {math.log(81, 3)}")

# 专用函数（精度更高）
print(f"\nlog2(1024)  = {math.log2(1024)}")
print(f"log10(1000) = {math.log10(1000)}")

# log1p：x接近0时的精确计算
x = 1e-15
print(f"\nlog(1 + {x})  = {math.log(1 + x)}")
print(f"log1p({x})    = {math.log1p(x)}")  # 更精确

# 输出：
# ln(e)  = 1.0
# ln(1)  = 0.0
# ln(10) = 2.302585
#
# log2(8)   = 3.0
# log10(100) = 2.0
# log3(81)  = 4.0
#
# log2(1024)  = 10.0
# log10(1000) = 3.0
#
# log(1 + 1e-15)  = 9.992007221626409e-16
# log1p(1e-15)    = 1.000000000000005e-15

import math

def entropy(probabilities):
    """计算信息熵（以2为底，单位：bits）"""
    return -sum(p * math.log2(p) for p in probabilities if p > 0)

def kl_divergence(p, q):
    """计算KL散度（相对熵）"""
    return sum(pi * math.log2(pi / qi) for pi, qi in zip(p, q) if pi > 0 and qi > 0)

# 公平硬币
coin = [0.5, 0.5]
print(f"硬币熵: {entropy(coin):.4f} bits")

# 骰子
dice = [1/6] * 6
print(f"骰子熵: {entropy(dice):.4f} bits")

# 不均匀硬币
biased = [0.9, 0.1]
print(f"偏硬币熵: {entropy(biased):.4f} bits")

# KL散度
p = [0.3, 0.4, 0.3]
q = [0.25, 0.5, 0.25]
print(f"\nKL(p||q) = {kl_divergence(p, q):.4f} bits")
print(f"KL(q||p) = {kl_divergence(q, p):.4f} bits")

# 输出：
# 硬币熵: 1.0000 bits
# 骰子熵: 2.5850 bits
# 偏硬币熵: 0.4690 bits
#
# KL(p||q) = 0.0386 bits
# KL(q||p) = 0.0414 bits

import math

# 分贝计算：dB = 10 * log10(P / P0)
power = 100
reference = 1
decibels = 10 * math.log10(power / reference)
print(f"功率比 {power}/{reference} = {decibels} dB")

# 声压级：SPL = 20 * log10(P / P0)
pressure = 2.0  # 帕斯卡
ref_pressure = 0.00002  # 参考声压
spl = 20 * math.log10(pressure / ref_pressure)
print(f"声压 {pressure}Pa 的声压级 = {spl:.1f} dB")

# pH值计算：pH = -log10[H+]
hydrogen_concentration = 1e-7  # 纯水
ph = -math.log10(hydrogen_concentration)
print(f"\n纯水 pH = {ph}")

# 酸性溶液
acid = 1e-3
print(f"酸性溶液 pH = {-math.log10(acid)}")

# 里氏震级：M = log10(A / A0)
amplitude = 10000
richter = math.log10(amplitude)
print(f"\n震级 = {richter:.1f}")

# 使用对数计算数字位数
def digit_count(n):
    """计算正整数的十进制位数"""
    return math.floor(math.log10(n)) + 1

print(f"\n12345 有 {digit_count(12345)} 位")
print(f"999 有 {digit_count(999)} 位")
print(f"1000 有 {digit_count(1000)} 位")

# 输出：
# 功率比 100/1 = 20.0 dB
# 声压 2.0Pa 的声压级 = 94.0 dB
#
# 纯水 pH = 7.0
# 酸性溶液 pH = 3.0
#
# 震级 = 4.0
#
# 12345 有 5 位
# 999 有 3 位
# 1000 有 4 位