Which is bigger, $15^{29}$ or $29^{15}$ ?

Q: Which is bigger, 15^29 or 29^15?

15^29 is bigger: 15^29 = 1.27834 × 10^34 and 29^15 = 8.62918 × 10^21.

$15^{29}$ is bigger: $15^{29}$ = 1.27834 × $10^{34}$ and $29^{15}$ = 8.62918 × $10^{21}$ .

15^{29} = 1.27834\times 10^{34}\ >\ 29^{15} = 8.62918\times 10^{21}

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The curve ln x / x, which decides every comparison of a^b with b^a. 15^29 against 29^15 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: $15^{29}$ = 12783403948858939111232757568359375 and $29^{15}$ = 8629188747598184440949.
2
So $15^{29}$ 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 15 and 29 are past e, where the function is decreasing, so the larger exponent loses and $15^{29}$ > $29^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 15^{29} = 1.27834\times 10^{34}\ >\ 29^{15} = 8.62918\times 10^{21} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

15^{29}

29^{15}

$15^{29}$ is bigger: $15^{29}$ = 1.27834 × $10^{34}$ and $29^{15}$ = 8.62918 × $10^{21}$ .

Which is bigger, 152915^{29}1529 or 291529^{15}2915?

How to work it out

Questions

Similar comparisons

Which is bigger, $15^{29}$ or $29^{15}$ ?