Which is bigger, $12^{15}$ or $15^{12}$ ?

Q: Which is bigger, 12^15 or 15^12?

12^15 is bigger: 12^15 = 1.54070 × 10^16 and 15^12 = 1.29746 × 10^14.

$12^{15}$ is bigger: $12^{15}$ = 1.54070 × $10^{16}$ and $15^{12}$ = 1.29746 × $10^{14}$ .

12^{15} = 1.54070\times 10^{16}\ >\ 15^{12} = 1.29746\times 10^{14}

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^15 against 15^12 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: $12^{15}$ = 15407021574586368 and $15^{12}$ = 129746337890625.
2
So $12^{15}$ 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 12 and 15 are past e, where the function is decreasing, so the larger exponent loses and $12^{15}$ > $15^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 15}{15}$

LaTeX source of the formulas on this page

\[ 12^{15} = 1.54070\times 10^{16}\ >\ 15^{12} = 1.29746\times 10^{14} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 15}{15} \]

Questions

Which is bigger,

12^{15}

15^{12}

$12^{15}$ is bigger: $12^{15}$ = 1.54070 × $10^{16}$ and $15^{12}$ = 1.29746 × $10^{14}$ .

Which is bigger, 121512^{15}1215 or 151215^{12}1512?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{15}$ or $15^{12}$ ?