Which is bigger, $17^{25}$ or $25^{17}$ ?

Q: Which is bigger, 17^25 or 25^17?

17^25 is bigger: 17^25 = 5.77062 × 10^30 and 25^17 = 5.82076 × 10^23.

$17^{25}$ is bigger: $17^{25}$ = 5.77062 × $10^{30}$ and $25^{17}$ = 5.82076 × $10^{23}$ .

17^{25} = 5.77062\times 10^{30}\ >\ 25^{17} = 5.82076\times 10^{23}

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The curve ln x / x, which decides every comparison of a^b with b^a. 17^25 against 25^17 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: $17^{25}$ = 5770627412348402378939569991057 and $25^{17}$ = 582076609134674072265625.
2
So $17^{25}$ 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 17 and 25 are past e, where the function is decreasing, so the larger exponent loses and $17^{25}$ > $25^{17}$ .
$\frac{\ln 17}{17}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 17^{25} = 5.77062\times 10^{30}\ >\ 25^{17} = 5.82076\times 10^{23} \]
\[ \frac{\ln 17}{17}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

17^{25}

25^{17}

$17^{25}$ is bigger: $17^{25}$ = 5.77062 × $10^{30}$ and $25^{17}$ = 5.82076 × $10^{23}$ .

Which is bigger, 172517^{25}1725 or 251725^{17}2517?

How to work it out

Questions

Similar comparisons

Which is bigger, $17^{25}$ or $25^{17}$ ?