Which is bigger, $7^{23}$ or $23^{7}$ ?

Q: Which is bigger, 7^23 or 23^7?

7^23 is bigger: 7^23 = 2.73687 × 10^19 and 23^7 = 3404825447.

$7^{23}$ is bigger: $7^{23}$ = 2.73687 × $10^{19}$ and $23^{7}$ = 3404825447.

7^{23} = 2.73687\times 10^{19}\ >\ 23^{7} = 3404825447

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^23 against 23^7 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: $7^{23}$ = 27368747340080916343 and $23^{7}$ = 3404825447.
2
So $7^{23}$ 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 23 are past e, where the function is decreasing, so the larger exponent loses and $7^{23}$ > $23^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 7^{23} = 2.73687\times 10^{19}\ >\ 23^{7} = 3404825447 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

7^{23}

23^{7}

$7^{23}$ is bigger: $7^{23}$ = 2.73687 × $10^{19}$ and $23^{7}$ = 3404825447.

Which is bigger, 7237^{23}723 or 23723^{7}237?

How to work it out

Questions

Similar comparisons

Which is bigger, $7^{23}$ or $23^{7}$ ?