Which is bigger, $15^{28}$ or $28^{15}$ ?

Q: Which is bigger, 15^28 or 28^15?

15^28 is bigger: 15^28 = 8.52226 × 10^32 and 28^15 = 5.09765 × 10^21.

$15^{28}$ is bigger: $15^{28}$ = 8.52226 × $10^{32}$ and $28^{15}$ = 5.09765 × $10^{21}$ .

15^{28} = 8.52226\times 10^{32}\ >\ 28^{15} = 5.09765\times 10^{21}

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The curve ln x / x, which decides every comparison of a^b with b^a. 15^28 against 28^15 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: $15^{28}$ = 852226929923929274082183837890625 and $28^{15}$ = 5097655355238390956032.
2
So $15^{28}$ 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 28 are past e, where the function is decreasing, so the larger exponent loses and $15^{28}$ > $28^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 15^{28} = 8.52226\times 10^{32}\ >\ 28^{15} = 5.09765\times 10^{21} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

15^{28}

28^{15}

$15^{28}$ is bigger: $15^{28}$ = 8.52226 × $10^{32}$ and $28^{15}$ = 5.09765 × $10^{21}$ .

Which is bigger, 152815^{28}1528 or 281528^{15}2815?

How to work it out

Questions

Similar comparisons

Which is bigger, $15^{28}$ or $28^{15}$ ?