Which is bigger, $20^{28}$ or $28^{20}$ ?

Q: Which is bigger, 20^28 or 28^20?

20^28 is bigger: 20^28 = 2.68435 × 10^36 and 28^20 = 8.77325 × 10^28.

$20^{28}$ is bigger: $20^{28}$ = 2.68435 × $10^{36}$ and $28^{20}$ = 8.77325 × $10^{28}$ .

20^{28} = 2.68435\times 10^{36}\ >\ 28^{20} = 8.77325\times 10^{28}

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The curve ln x / x, which decides every comparison of a^b with b^a. 20^28 against 28^20 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: $20^{28}$ = 2684354560000000000000000000000000000 and $28^{20}$ = 87732524600823436081182539776.
2
So $20^{28}$ 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 28 are past e, where the function is decreasing, so the larger exponent loses and $20^{28}$ > $28^{20}$ .
$\frac{\ln 20}{20}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 20^{28} = 2.68435\times 10^{36}\ >\ 28^{20} = 8.77325\times 10^{28} \]
\[ \frac{\ln 20}{20}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

20^{28}

28^{20}

$20^{28}$ is bigger: $20^{28}$ = 2.68435 × $10^{36}$ and $28^{20}$ = 8.77325 × $10^{28}$ .

Which is bigger, 202820^{28}2028 or 282028^{20}2820?

How to work it out

Questions

Similar comparisons

Which is bigger, $20^{28}$ or $28^{20}$ ?