Which is bigger, $13^{15}$ or $15^{13}$ ?

Q: Which is bigger, 13^15 or 15^13?

13^15 is bigger: 13^15 = 5.11858 × 10^16 and 15^13 = 1.94619 × 10^15.

$13^{15}$ is bigger: $13^{15}$ = 5.11858 × $10^{16}$ and $15^{13}$ = 1.94619 × $10^{15}$ .

13^{15} = 5.11858\times 10^{16}\ >\ 15^{13} = 1.94619\times 10^{15}

Explain it step by step in chat Make a short video of it

The curve ln x / x, which decides every comparison of a^b with b^a. 13^15 against 15^13 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: $13^{15}$ = 51185893014090757 and $15^{13}$ = 1946195068359375.
2
So $13^{15}$ 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 13 and 15 are past e, where the function is decreasing, so the larger exponent loses and $13^{15}$ > $15^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 15}{15}$

LaTeX source of the formulas on this page

\[ 13^{15} = 5.11858\times 10^{16}\ >\ 15^{13} = 1.94619\times 10^{15} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 15}{15} \]

Questions

Which is bigger,

13^{15}

15^{13}

$13^{15}$ is bigger: $13^{15}$ = 5.11858 × $10^{16}$ and $15^{13}$ = 1.94619 × $10^{15}$ .

Which is bigger, 131513^{15}1315 or 151315^{13}1513?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{15}$ or $15^{13}$ ?