Which is bigger, $22^{27}$ or $27^{22}$ ?

Q: Which is bigger, 22^27 or 27^22?

22^27 is bigger: 22^27 = 1.75959 × 10^36 and 27^22 = 3.09031 × 10^31.

$22^{27}$ is bigger: $22^{27}$ = 1.75959 × $10^{36}$ and $27^{22}$ = 3.09031 × $10^{31}$ .

22^{27} = 1.75959\times 10^{36}\ >\ 27^{22} = 3.09031\times 10^{31}

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The curve ln x / x, which decides every comparison of a^b with b^a. 22^27 against 27^22 is the same question as which of the two marked points is higher; the curve peaks at x = e.

How to work it out

1
Compute both exactly: $22^{27}$ = 1759593634476317566025195247357657088 and $27^{22}$ = 30903154382632612361920641803529.
2
So $22^{27}$ is bigger.
3
The reason: taking logs, $a^{b}$ against $b^{a}$ is the same as $\frac{\ln a}{a}$ against $\frac{\ln b}{b}$ . The function $\frac{\ln x}{x}$ rises until $x = e \approx 2.718$ and falls afterwards. Both 22 and 27 are past e, where the function is decreasing, so the larger exponent loses and $22^{27}$ > $27^{22}$ .
$\frac{\ln 22}{22}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 22^{27} = 1.75959\times 10^{36}\ >\ 27^{22} = 3.09031\times 10^{31} \]
\[ \frac{\ln 22}{22}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

22^{27}

27^{22}

$22^{27}$ is bigger: $22^{27}$ = 1.75959 × $10^{36}$ and $27^{22}$ = 3.09031 × $10^{31}$ .

Which is bigger, 222722^{27}2227 or 272227^{22}2722?

How to work it out

Questions

Similar comparisons

Which is bigger, $22^{27}$ or $27^{22}$ ?