Which is bigger, $20^{27}$ or $27^{20}$ ?

Q: Which is bigger, 20^27 or 27^20?

20^27 is bigger: 20^27 = 1.34217 × 10^35 and 27^20 = 4.23911 × 10^28.

$20^{27}$ is bigger: $20^{27}$ = 1.34217 × $10^{35}$ and $27^{20}$ = 4.23911 × $10^{28}$ .

20^{27} = 1.34217\times 10^{35}\ >\ 27^{20} = 4.23911\times 10^{28}

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The curve ln x / x, which decides every comparison of a^b with b^a. 20^27 against 27^20 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: $20^{27}$ = 134217728000000000000000000000000000 and $27^{20}$ = 42391158275216203514294433201.
2
So $20^{27}$ 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 20 and 27 are past e, where the function is decreasing, so the larger exponent loses and $20^{27}$ > $27^{20}$ .
$\frac{\ln 20}{20}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 20^{27} = 1.34217\times 10^{35}\ >\ 27^{20} = 4.23911\times 10^{28} \]
\[ \frac{\ln 20}{20}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

20^{27}

27^{20}

$20^{27}$ is bigger: $20^{27}$ = 1.34217 × $10^{35}$ and $27^{20}$ = 4.23911 × $10^{28}$ .

Which is bigger, 202720^{27}2027 or 272027^{20}2720?

How to work it out

Questions

Similar comparisons

Which is bigger, $20^{27}$ or $27^{20}$ ?