Which is bigger, $7^{29}$ or $29^{7}$ ?

Q: Which is bigger, 7^29 or 29^7?

7^29 is bigger: 7^29 = 3.21990 × 10^24 and 29^7 = 17249876309.

$7^{29}$ is bigger: $7^{29}$ = 3.21990 × $10^{24}$ and $29^{7}$ = 17249876309.

7^{29} = 3.21990\times 10^{24}\ >\ 29^{7} = 17249876309

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^29 against 29^7 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: $7^{29}$ = 3219905755813179726837607 and $29^{7}$ = 17249876309.
2
So $7^{29}$ 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 7 and 29 are past e, where the function is decreasing, so the larger exponent loses and $7^{29}$ > $29^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 7^{29} = 3.21990\times 10^{24}\ >\ 29^{7} = 17249876309 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

7^{29}

29^{7}

$7^{29}$ is bigger: $7^{29}$ = 3.21990 × $10^{24}$ and $29^{7}$ = 17249876309.

Which is bigger, 7297^{29}729 or 29729^{7}297?

How to work it out

Questions

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Which is bigger, $7^{29}$ or $29^{7}$ ?