Which is bigger, $15^{20}$ or $20^{15}$ ?

Q: Which is bigger, 15^20 or 20^15?

15^20 is bigger: 15^20 = 3.32525 × 10^23 and 20^15 = 3.27680 × 10^19.

$15^{20}$ is bigger: $15^{20}$ = 3.32525 × $10^{23}$ and $20^{15}$ = 3.27680 × $10^{19}$ .

15^{20} = 3.32525\times 10^{23}\ >\ 20^{15} = 3.27680\times 10^{19}

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The curve ln x / x, which decides every comparison of a^b with b^a. 15^20 against 20^15 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: $15^{20}$ = 332525673007965087890625 and $20^{15}$ = 32768000000000000000.
2
So $15^{20}$ 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 15 and 20 are past e, where the function is decreasing, so the larger exponent loses and $15^{20}$ > $20^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 15^{20} = 3.32525\times 10^{23}\ >\ 20^{15} = 3.27680\times 10^{19} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

15^{20}

20^{15}

$15^{20}$ is bigger: $15^{20}$ = 3.32525 × $10^{23}$ and $20^{15}$ = 3.27680 × $10^{19}$ .

Which is bigger, 152015^{20}1520 or 201520^{15}2015?

How to work it out

Questions

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Which is bigger, $15^{20}$ or $20^{15}$ ?