Which is bigger, $16^{27}$ or $27^{16}$ ?

Q: Which is bigger, 16^27 or 27^16?

16^27 is bigger: 16^27 = 3.24518 × 10^32 and 27^16 = 7.97664 × 10^22.

$16^{27}$ is bigger: $16^{27}$ = 3.24518 × $10^{32}$ and $27^{16}$ = 7.97664 × $10^{22}$ .

16^{27} = 3.24518\times 10^{32}\ >\ 27^{16} = 7.97664\times 10^{22}

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The curve ln x / x, which decides every comparison of a^b with b^a. 16^27 against 27^16 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: $16^{27}$ = 324518553658426726783156020576256 and $27^{16}$ = 79766443076872509863361.
2
So $16^{27}$ 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 16 and 27 are past e, where the function is decreasing, so the larger exponent loses and $16^{27}$ > $27^{16}$ .
$\frac{\ln 16}{16}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 16^{27} = 3.24518\times 10^{32}\ >\ 27^{16} = 7.97664\times 10^{22} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

16^{27}

27^{16}

$16^{27}$ is bigger: $16^{27}$ = 3.24518 × $10^{32}$ and $27^{16}$ = 7.97664 × $10^{22}$ .

Which is bigger, 162716^{27}1627 or 271627^{16}2716?

How to work it out

Questions

Similar comparisons

Which is bigger, $16^{27}$ or $27^{16}$ ?