So I've stumbled upon this problem:
Compare the numbers:
$$9^{8^{8^9}} \text{ and }\,8^{9^{9^8}}$$
I got this in a test and I had no idea what the answer to the problem is... Can someone give me the answer and how you can find it? It's obviously not possible to calculate such huge numbers, so there's probably another way to find it which I haven't figured out.
On
On my conclusion, $$ 9^{8^{8^9}} > 8^{9^{9^8}} $$
To explain this,
Manually we know, $$ 9\ln(8) > 8\ln(9) $$
therefore, $$ 8^9 > 9^8 $$ $$ 8^9 \approx 3(9^8) $$
now $\ln(9)=2.19$
$\ln(8)=2.07$
i.e. $ \frac{\ln(9)}{\ln(8)}=1.05 $,
therefore $$ 8^9\ln(8) > 9^8\ln(9)$$
$$ 8^{8^9} > 9^{9^8} $$
now, $$ 8^{8^9}\ln(9) > 9^{9^8}\ln(8) $$
concluding,
$$ 9^{8^{8^9}} > 8^{9^{9^8}} $$
$8^9 > 9^8$
In fact it is more than $3$ times greater.
Which would suggest that $8^{8^9} \gg 9^{9^8}$ ($\gg$ means significantly greater)
So then the next level
$9^{8^{8^9}}$ is a larger base to a larger exponent than $8^{9^{9^8}}$