Which is bigger, $15^{17}$ or $17^{15}$ ?

Q: Which is bigger, 15^17 or 17^15?

15^17 is bigger: 15^17 = 9.85261 × 10^19 and 17^15 = 2.86242 × 10^18.

$15^{17}$ is bigger: $15^{17}$ = 9.85261 × $10^{19}$ and $17^{15}$ = 2.86242 × $10^{18}$ .

15^{17} = 9.85261\times 10^{19}\ >\ 17^{15} = 2.86242\times 10^{18}

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The curve ln x / x, which decides every comparison of a^b with b^a. 15^17 against 17^15 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: $15^{17}$ = 98526125335693359375 and $17^{15}$ = 2862423051509815793.
2
So $15^{17}$ 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 15 and 17 are past e, where the function is decreasing, so the larger exponent loses and $15^{17}$ > $17^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 17}{17}$

LaTeX source of the formulas on this page

\[ 15^{17} = 9.85261\times 10^{19}\ >\ 17^{15} = 2.86242\times 10^{18} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 17}{17} \]

Questions

Which is bigger,

15^{17}

17^{15}

$15^{17}$ is bigger: $15^{17}$ = 9.85261 × $10^{19}$ and $17^{15}$ = 2.86242 × $10^{18}$ .

Which is bigger, 151715^{17}1517 or 171517^{15}1715?

How to work it out

Questions

Similar comparisons

Which is bigger, $15^{17}$ or $17^{15}$ ?