## Project Euler Problem 99 Solution

## Question

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

However, confirming that $632382^{518061} \gt 519432^{525806}$ 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.

## Commentary

As the question notes, it would be difficult to evaluate these exponents directly as they contain over three million digits each. Therefore, we must find a way to reduce these exponents to a manageable size.

To do so, we use the identity $\log{a^x} = x \times \log{a}$. Since $\log{b} \gt \log{a}$ if $ b a$, this will serve as an approximation of calculating the exponents directly.

## Python

```
#!/usr/bin/env python
import os
from math import log10
largest = [0, 0]
for i, line in enumerate(open(os.path.join(os.path.dirname(__file__), 'base_exp.txt'))):
a, x = list(map(int, line.split(',')))
if x * log10(a) > largest[0]:
largest = [x * log10(a), i+1]
print(largest[1])
```

```
$ time python3 largest-exponent.py
real 0m0.025s
user 0m0.020s
sys 0m0.004s
```

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