Which is bigger, $8^{23}$ or $23^{8}$ ?

Q: Which is bigger, 8^23 or 23^8?

8^23 is bigger: 8^23 = 5.90295 × 10^20 and 23^8 = 78310985281.

$8^{23}$ is bigger: $8^{23}$ = 5.90295 × $10^{20}$ and $23^{8}$ = 78310985281.

8^{23} = 5.90295\times 10^{20}\ >\ 23^{8} = 78310985281

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The curve ln x / x, which decides every comparison of a^b with b^a. 8^23 against 23^8 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: $8^{23}$ = 590295810358705651712 and $23^{8}$ = 78310985281.
2
So $8^{23}$ 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 8 and 23 are past e, where the function is decreasing, so the larger exponent loses and $8^{23}$ > $23^{8}$ .
$\frac{\ln 8}{8}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 8^{23} = 5.90295\times 10^{20}\ >\ 23^{8} = 78310985281 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

8^{23}

23^{8}

$8^{23}$ is bigger: $8^{23}$ = 5.90295 × $10^{20}$ and $23^{8}$ = 78310985281.

Which is bigger, 8238^{23}823 or 23823^{8}238?

How to work it out

Questions

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Which is bigger, $8^{23}$ or $23^{8}$ ?