Here is another argument using power-series-based computations of log, but slightly different in its details to Ivans.
Kayla Kline
Beginner2022-01-22Added 37 answers
Note that
Hence we need to prove
Now,
I claim that and that . Then
proving the desired result.
Since you explicitly ask not to use a computer let me show the two claimed inequalities by hand:
Finally, since 80^4=5^4\cdot2^{16} we only need to show that . But
RizerMix
Expert2022-01-27Added 656 answers
I will use approximations by power series. It is easy to show that the inequality is equivalent to: On the other hand consider the power series expansion for around .It is:
For we get:
On the other hand for the linear term gives:
. Now we aim to prove the following:
The first and last are proved already by the approximations. We need the middle. It is equivalent to the positivity of:
The last is positive because