Which is bigger, $4^{27}$ or $27^{4}$ ?

Q: Which is bigger, 4^27 or 27^4?

4^27 is bigger: 4^27 = 1.80143 × 10^16 and 27^4 = 531441.

$4^{27}$ is bigger: $4^{27}$ = 1.80143 × $10^{16}$ and $27^{4}$ = 531441.

4^{27} = 1.80143\times 10^{16}\ >\ 27^{4} = 531441

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^27 against 27^4 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: $4^{27}$ = 18014398509481984 and $27^{4}$ = 531441.
2
So $4^{27}$ 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 4 and 27 are past e, where the function is decreasing, so the larger exponent loses and $4^{27}$ > $27^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 4^{27} = 1.80143\times 10^{16}\ >\ 27^{4} = 531441 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

4^{27}

27^{4}

$4^{27}$ is bigger: $4^{27}$ = 1.80143 × $10^{16}$ and $27^{4}$ = 531441.

Which is bigger, 4274^{27}427 or 27427^{4}274?

How to work it out

Questions

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Which is bigger, $4^{27}$ or $27^{4}$ ?