Which is bigger, $25^{27}$ or $27^{25}$ ?

Q: Which is bigger, 25^27 or 27^25?

25^27 is bigger: 25^27 = 5.55111 × 10^37 and 27^25 = 6.08266 × 10^35.

$25^{27}$ is bigger: $25^{27}$ = 5.55111 × $10^{37}$ and $27^{25}$ = 6.08266 × $10^{35}$ .

25^{27} = 5.55111\times 10^{37}\ >\ 27^{25} = 6.08266\times 10^{35}

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The curve ln x / x, which decides every comparison of a^b with b^a. 25^27 against 27^25 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: $25^{27}$ = 55511151231257827021181583404541015625 and $27^{25}$ = 608266787713357709119683992618861307.
2
So $25^{27}$ 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 25 and 27 are past e, where the function is decreasing, so the larger exponent loses and $25^{27}$ > $27^{25}$ .
$\frac{\ln 25}{25}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 25^{27} = 5.55111\times 10^{37}\ >\ 27^{25} = 6.08266\times 10^{35} \]
\[ \frac{\ln 25}{25}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

25^{27}

27^{25}

$25^{27}$ is bigger: $25^{27}$ = 5.55111 × $10^{37}$ and $27^{25}$ = 6.08266 × $10^{35}$ .

Which is bigger, 252725^{27}2527 or 272527^{25}2725?

How to work it out

Questions

Similar comparisons

Which is bigger, $25^{27}$ or $27^{25}$ ?