Project Euler – Problem 99 Solution

Problem

Comparing two numbers written in index form like 2¹¹ and 3⁷ is not difficult, as any calculator would confirm that 2¹¹ = 2048 < 3⁷ = 2187.

However, confirming that 632382⁵¹⁸⁰⁶¹ > 519432⁵²⁵⁸⁰⁶ 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.

Solution

This question looks tricky at first as working out the actual value of each of the base/exponent pair in the file can take a looong time.. but as many others have suggested, you can compare the logarithm of the actual value instead, which can be expressed in the following form:

which can be computed very quickly and the rest is easy.