Which is bigger, $7^{15}$ or $15^{7}$ ?

Q: Which is bigger, 7^15 or 15^7?

7^15 is bigger: 7^15 = 4.74756 × 10^12 and 15^7 = 170859375.

$7^{15}$ is bigger: $7^{15}$ = 4.74756 × $10^{12}$ and $15^{7}$ = 170859375.

7^{15} = 4747561509943\ >\ 15^{7} = 170859375

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^15 against 15^7 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: $7^{15}$ = 4747561509943 and $15^{7}$ = 170859375.
2
So $7^{15}$ 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 7 and 15 are past e, where the function is decreasing, so the larger exponent loses and $7^{15}$ > $15^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 15}{15}$

LaTeX source of the formulas on this page

\[ 7^{15} = 4747561509943\ >\ 15^{7} = 170859375 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 15}{15} \]

Questions

Which is bigger,

7^{15}

15^{7}

$7^{15}$ is bigger: $7^{15}$ = 4.74756 × $10^{12}$ and $15^{7}$ = 170859375.

Which is bigger, 7157^{15}715 or 15715^{7}157?

How to work it out

Questions

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Which is bigger, $7^{15}$ or $15^{7}$ ?