Which is bigger, $7^{27}$ or $27^{7}$ ?

Q: Which is bigger, 7^27 or 27^7?

7^27 is bigger: 7^27 = 6.57123 × 10^22 and 27^7 = 10460353203.

$7^{27}$ is bigger: $7^{27}$ = 6.57123 × $10^{22}$ and $27^{7}$ = 10460353203.

7^{27} = 6.57123\times 10^{22}\ >\ 27^{7} = 10460353203

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^27 against 27^7 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: $7^{27}$ = 65712362363534280139543 and $27^{7}$ = 10460353203.
2
So $7^{27}$ 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 7 and 27 are past e, where the function is decreasing, so the larger exponent loses and $7^{27}$ > $27^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 7^{27} = 6.57123\times 10^{22}\ >\ 27^{7} = 10460353203 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

7^{27}

27^{7}

$7^{27}$ is bigger: $7^{27}$ = 6.57123 × $10^{22}$ and $27^{7}$ = 10460353203.

Which is bigger, 7277^{27}727 or 27727^{7}277?

How to work it out

Questions

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Which is bigger, $7^{27}$ or $27^{7}$ ?