Which is bigger, $7^{12}$ or $12^{7}$ ?

Q: Which is bigger, 7^12 or 12^7?

7^12 is bigger: 7^12 = 13841287201 and 12^7 = 35831808.

$7^{12}$ is bigger: $7^{12}$ = 13841287201 and $12^{7}$ = 35831808.

7^{12} = 13841287201\ >\ 12^{7} = 35831808

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^12 against 12^7 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: $7^{12}$ = 13841287201 and $12^{7}$ = 35831808.
2
So $7^{12}$ 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 12 are past e, where the function is decreasing, so the larger exponent loses and $7^{12}$ > $12^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 12}{12}$

LaTeX source of the formulas on this page

\[ 7^{12} = 13841287201\ >\ 12^{7} = 35831808 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 12}{12} \]

Questions

Which is bigger,

7^{12}

12^{7}

$7^{12}$ is bigger: $7^{12}$ = 13841287201 and $12^{7}$ = 35831808.

Which is bigger, 7127^{12}712 or 12712^{7}127?

How to work it out

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Which is bigger, $7^{12}$ or $12^{7}$ ?