Project Euler 99

题目

Largest exponential

Comparing two numbers written in index form like \(2^{11}\) and \(3^7\) is not difficult, as any calculator would confirm that \(2^{11} = 2048 < 3^7 = 2187\).

However, confirming that \(632382^{518061} > 519432^{525806}\) would be much more difficult, as both numbers contain over three million digits.

Using base_exp.txt (right click and ‘Save Link/Target As…’), a 22K text file containing one thousand lines with a base/exponent pair on each line, determine which line number has the greatest numerical value. NOTE: The first two lines in the file represent the numbers in the example given above.

解决方案

由于对数函数\(y=\ln x\) 在 \((0,+\infty)\)上递增。因此我们可以这些形如\(a^b\)的所有数全部取对数化成\(b\ln a\)，它们的相对大小依然不变。

因此，直接将所有数的对数值计算出即可完成比较。

代码

from math import log

ls = open('p099_base_exp.txt', 'r').readlines()
ans, i, mx = 0, 0, 0
for s in ls:
    i += 1
    s = s[:-1]
    p, e = [int(x) for x in s.split(',')]
    w = log(p) * e
    if w > mx:
        mx = w
        ans = i
print(ans)