Which is bigger, $20^{30}$ or $30^{20}$ ?

Q: Which is bigger, 20^30 or 30^20?

20^30 is bigger: 20^30 = 1.07374 × 10^39 and 30^20 = 3.48678 × 10^29.

$20^{30}$ is bigger: $20^{30}$ = 1.07374 × $10^{39}$ and $30^{20}$ = 3.48678 × $10^{29}$ .

20^{30} = 1.07374\times 10^{39}\ >\ 30^{20} = 3.48678\times 10^{29}

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The curve ln x / x, which decides every comparison of a^b with b^a. 20^30 against 30^20 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: $20^{30}$ = 1073741824000000000000000000000000000000 and $30^{20}$ = 348678440100000000000000000000.
2
So $20^{30}$ 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 30 are past e, where the function is decreasing, so the larger exponent loses and $20^{30}$ > $30^{20}$ .
$\frac{\ln 20}{20}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 20^{30} = 1.07374\times 10^{39}\ >\ 30^{20} = 3.48678\times 10^{29} \]
\[ \frac{\ln 20}{20}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

20^{30}

30^{20}

$20^{30}$ is bigger: $20^{30}$ = 1.07374 × $10^{39}$ and $30^{20}$ = 3.48678 × $10^{29}$ .

Which is bigger, 203020^{30}2030 or 302030^{20}3020?

How to work it out

Questions

Similar comparisons

Which is bigger, $20^{30}$ or $30^{20}$ ?