Which is bigger, $27^{29}$ or $29^{27}$ ?

Q: Which is bigger, 27^29 or 29^27?

27^29 is bigger: 27^29 = 3.23257 × 10^41 and 29^27 = 3.05313 × 10^39.

$27^{29}$ is bigger: $27^{29}$ = 3.23257 × $10^{41}$ and $29^{27}$ = 3.05313 × $10^{39}$ .

27^{29} = 3.23257\times 10^{41}\ >\ 29^{27} = 3.05313\times 10^{39}

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The curve ln x / x, which decides every comparison of a^b with b^a. 27^29 against 29^27 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: $27^{29}$ = 323257909929174534292273980721360271853387 and $29^{27}$ = 3053134545970524535745336759489912159909.
2
So $27^{29}$ 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 27 and 29 are past e, where the function is decreasing, so the larger exponent loses and $27^{29}$ > $29^{27}$ .
$\frac{\ln 27}{27}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 27^{29} = 3.23257\times 10^{41}\ >\ 29^{27} = 3.05313\times 10^{39} \]
\[ \frac{\ln 27}{27}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

27^{29}

29^{27}

$27^{29}$ is bigger: $27^{29}$ = 3.23257 × $10^{41}$ and $29^{27}$ = 3.05313 × $10^{39}$ .

Which is bigger, 272927^{29}2729 or 292729^{27}2927?

How to work it out

Questions

Similar comparisons

Which is bigger, $27^{29}$ or $29^{27}$ ?