Which is bigger, $15^{24}$ or $24^{15}$ ?

Q: Which is bigger, 15^24 or 24^15?

15^24 is bigger: 15^24 = 1.68341 × 10^28 and 24^15 = 5.04857 × 10^20.

$15^{24}$ is bigger: $15^{24}$ = 1.68341 × $10^{28}$ and $24^{15}$ = 5.04857 × $10^{20}$ .

15^{24} = 1.68341\times 10^{28}\ >\ 24^{15} = 5.04857\times 10^{20}

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The curve ln x / x, which decides every comparison of a^b with b^a. 15^24 against 24^15 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: $15^{24}$ = 16834112196028232574462890625 and $24^{15}$ = 504857282956046106624.
2
So $15^{24}$ 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 24 are past e, where the function is decreasing, so the larger exponent loses and $15^{24}$ > $24^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 15^{24} = 1.68341\times 10^{28}\ >\ 24^{15} = 5.04857\times 10^{20} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

15^{24}

24^{15}

$15^{24}$ is bigger: $15^{24}$ = 1.68341 × $10^{28}$ and $24^{15}$ = 5.04857 × $10^{20}$ .

Which is bigger, 152415^{24}1524 or 241524^{15}2415?

How to work it out

Questions

Similar comparisons

Which is bigger, $15^{24}$ or $24^{15}$ ?