## Comparing Logs

What is greater, or ?

The trick here is to compare both to rather than to each other. Start by comparing to . Function is monotonic so we can apply it to both sides to get

Function is also monotonic so we can apply it to both sides

and see that

therefore

Now compare to . Function is monotonic so we can apply it to both sides

Function is also monotonic so we can apply it to both sides

therefore

Finally

Source: problem 17 in https://arxiv.org/pdf/1110.1556v2.pdf