Which is bigger, $24^{26}$ or $26^{24}$ ?

Q: Which is bigger, 24^26 or 26^24?

24^26 is bigger: 24^26 = 7.68231 × 10^35 and 26^24 = 9.10668 × 10^33.

$24^{26}$ is bigger: $24^{26}$ = 7.68231 × $10^{35}$ and $26^{24}$ = 9.10668 × $10^{33}$ .

24^{26} = 7.68231\times 10^{35}\ >\ 26^{24} = 9.10668\times 10^{33}

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The curve ln x / x, which decides every comparison of a^b with b^a. 24^26 against 26^24 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: $24^{26}$ = 768231807465763655682670928358014976 and $26^{24}$ = 9106685769537214956799814036094976.
2
So $24^{26}$ 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 24 and 26 are past e, where the function is decreasing, so the larger exponent loses and $24^{26}$ > $26^{24}$ .
$\frac{\ln 24}{24}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 24^{26} = 7.68231\times 10^{35}\ >\ 26^{24} = 9.10668\times 10^{33} \]
\[ \frac{\ln 24}{24}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

24^{26}

26^{24}

$24^{26}$ is bigger: $24^{26}$ = 7.68231 × $10^{35}$ and $26^{24}$ = 9.10668 × $10^{33}$ .

Which is bigger, 242624^{26}2426 or 262426^{24}2624?

How to work it out

Questions

Similar comparisons

Which is bigger, $24^{26}$ or $26^{24}$ ?