Linked Questions
26 questions linked to/from Fastest way to check if $x^y > y^x$?
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Whether $2^{38}$ or $3^{33}$ is greater without needing a calculator [closed]
My question is about figuring out whether $2^{38}$ or $3^{33}$ is greater without needing a calculator, by using the Mobius function or by other means?
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3
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Unspecified $x^y$ vs. $y^x$ - which is larger?
Given only the expressions $x^y$ and $y^x$ and no additional information except $x\neq y$ (and the meta-knowledge that the problem was presented in the context of induction), is it possible to ...
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3
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Compare two below natural numbers: $2016^{2017} < 2017^{2016}$ [closed]
Help me Compare the two following natural numbers below
$$2016^{2017} < 2017^{2016}?$$
Many thanks.
3
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1
answer
346
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Collection of Non-Trick Questions That Require Work to Answer
This question is inspired by a comment on a recent popular question:
Which area is larger, the blue area, or the white area?
Warning: Spoiler Below! - Don't keep reading if you want to solve this ...
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3
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Showing that $2^{50}<3^{33}$ [duplicate]
I'd never encountered a problem like "show that $2^{50}<3^{33}$" but I think I ended up solving it after doing some weird stuff with logs and Maclaurin series:
$$
\begin{align*}
2^{50}&<3^{...
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3
votes
1
answer
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Which is larger, $e^\pi$ or $\pi^e$? [duplicate]
I don't know how to approach this.
I tried expanding $e^{\pi}$ using the power series but that was a dead end since I didn't know what to do with it. I tried estimating if $e \log({\pi})$ was ...
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1
vote
1
answer
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How to compare $\sqrt{2}^{e^\pi}$ and $\pi^{e^\sqrt{2}}$?
I know there are nice ways to compare $x^y$, $y^x$ such as here. And for multiple exponential, I think the problem becomes hard. For instance, I want to determine which of $x^{y^z}$ and $z^{y^x}$ is ...
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Prove that $(n+3)^2 \le 2^{n+3} , n\in\mathbb{N}$ without induction.
Prove that: $$(n+3)^2 \le 2^{n+3},\quad n\in\mathbb{N}$$
Please show me how to prove this inequality using a method other than mathematical induction. I was solving some questions based on the ...
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votes
1
answer
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SMO - 2010: finding least or greatest number
Among the five real numbers below, which one is the smallest?
$\text{(A)}\ \ \sqrt[\leftroot{-2}\uproot{2}2009]{2010};\quad\text{(B)}\ \ \sqrt[\leftroot{-2}\uproot{2}2010]{2009};\quad\text{(C)}\ \ ...
user880107
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Proving an exponential inequality using calculus
Prove ( using calculus) that $20.17^{20.16}<20.16^{20.17}$.
How do I do that?
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A question about exponential functions: $a^b>b^a$ for $b>a>e$ [closed]
How can we prove that for $b>a>e$ ($e$ being the Euler’s number), $a^b$ is greater than $b^a$?
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