## 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.