Which is bigger, $17^{26}$ or $26^{17}$ ?

Q: Which is bigger, 17^26 or 26^17?

17^26 is bigger: 17^26 = 9.81006 × 10^31 and 26^17 = 1.13382 × 10^24.

$17^{26}$ is bigger: $17^{26}$ = 9.81006 × $10^{31}$ and $26^{17}$ = 1.13382 × $10^{24}$ .

17^{26} = 9.81006\times 10^{31}\ >\ 26^{17} = 1.13382\times 10^{24}

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The curve ln x / x, which decides every comparison of a^b with b^a. 17^26 against 26^17 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: $17^{26}$ = 98100666009922840441972689847969 and $26^{17}$ = 1133827315385150725554176.
2
So $17^{26}$ 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 17 and 26 are past e, where the function is decreasing, so the larger exponent loses and $17^{26}$ > $26^{17}$ .
$\frac{\ln 17}{17}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 17^{26} = 9.81006\times 10^{31}\ >\ 26^{17} = 1.13382\times 10^{24} \]
\[ \frac{\ln 17}{17}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

17^{26}

26^{17}

$17^{26}$ is bigger: $17^{26}$ = 9.81006 × $10^{31}$ and $26^{17}$ = 1.13382 × $10^{24}$ .

Which is bigger, 172617^{26}1726 or 261726^{17}2617?

How to work it out

Questions

Similar comparisons

Which is bigger, $17^{26}$ or $26^{17}$ ?