Which is bigger, $15^{27}$ or $27^{15}$ ?

Q: Which is bigger, 15^27 or 27^15?

15^27 is bigger: 15^27 = 5.68151 × 10^31 and 27^15 = 2.95431 × 10^21.

$15^{27}$ is bigger: $15^{27}$ = 5.68151 × $10^{31}$ and $27^{15}$ = 2.95431 × $10^{21}$ .

15^{27} = 5.68151\times 10^{31}\ >\ 27^{15} = 2.95431\times 10^{21}

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The curve ln x / x, which decides every comparison of a^b with b^a. 15^27 against 27^15 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: $15^{27}$ = 56815128661595284938812255859375 and $27^{15}$ = 2954312706550833698643.
2
So $15^{27}$ 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 15 and 27 are past e, where the function is decreasing, so the larger exponent loses and $15^{27}$ > $27^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 15^{27} = 5.68151\times 10^{31}\ >\ 27^{15} = 2.95431\times 10^{21} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

15^{27}

27^{15}

$15^{27}$ is bigger: $15^{27}$ = 5.68151 × $10^{31}$ and $27^{15}$ = 2.95431 × $10^{21}$ .

Which is bigger, 152715^{27}1527 or 271527^{15}2715?

How to work it out

Questions

Similar comparisons

Which is bigger, $15^{27}$ or $27^{15}$ ?