Which is bigger, $14^{26}$ or $26^{14}$ ?

Q: Which is bigger, 14^26 or 26^14?

14^26 is bigger: 14^26 = 6.29983 × 10^29 and 26^14 = 6.45099 × 10^19.

$14^{26}$ is bigger: $14^{26}$ = 6.29983 × $10^{29}$ and $26^{14}$ = 6.45099 × $10^{19}$ .

14^{26} = 6.29983\times 10^{29}\ >\ 26^{14} = 6.45099\times 10^{19}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^26 against 26^14 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: $14^{26}$ = 629983141281877223603213172736 and $26^{14}$ = 64509974703297150976.
2
So $14^{26}$ 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 14 and 26 are past e, where the function is decreasing, so the larger exponent loses and $14^{26}$ > $26^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 14^{26} = 6.29983\times 10^{29}\ >\ 26^{14} = 6.45099\times 10^{19} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

14^{26}

26^{14}

$14^{26}$ is bigger: $14^{26}$ = 6.29983 × $10^{29}$ and $26^{14}$ = 6.45099 × $10^{19}$ .

Which is bigger, 142614^{26}1426 or 261426^{14}2614?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{26}$ or $26^{14}$ ?