Which is bigger, $15^{21}$ or $21^{15}$ ?

Q: Which is bigger, 15^21 or 21^15?

15^21 is bigger: 15^21 = 4.98788 × 10^24 and 21^15 = 6.81223 × 10^19.

$15^{21}$ is bigger: $15^{21}$ = 4.98788 × $10^{24}$ and $21^{15}$ = 6.81223 × $10^{19}$ .

15^{21} = 4.98788\times 10^{24}\ >\ 21^{15} = 6.81223\times 10^{19}

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The curve ln x / x, which decides every comparison of a^b with b^a. 15^21 against 21^15 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: $15^{21}$ = 4987885095119476318359375 and $21^{15}$ = 68122318582951682301.
2
So $15^{21}$ 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 21 are past e, where the function is decreasing, so the larger exponent loses and $15^{21}$ > $21^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 21}{21}$

LaTeX source of the formulas on this page

\[ 15^{21} = 4.98788\times 10^{24}\ >\ 21^{15} = 6.81223\times 10^{19} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger,

15^{21}

21^{15}

$15^{21}$ is bigger: $15^{21}$ = 4.98788 × $10^{24}$ and $21^{15}$ = 6.81223 × $10^{19}$ .

Which is bigger, 152115^{21}1521 or 211521^{15}2115?

How to work it out

Questions

Similar comparisons

Which is bigger, $15^{21}$ or $21^{15}$ ?