Which is bigger, $12^{27}$ or $27^{12}$ ?

Q: Which is bigger, 12^27 or 27^12?

12^27 is bigger: 12^27 = 1.37370 × 10^29 and 27^12 = 1.50094 × 10^17.

$12^{27}$ is bigger: $12^{27}$ = 1.37370 × $10^{29}$ and $27^{12}$ = 1.50094 × $10^{17}$ .

12^{27} = 1.37370\times 10^{29}\ >\ 27^{12} = 1.50094\times 10^{17}

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^27 against 27^12 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: $12^{27}$ = 137370551967459378662586974208 and $27^{12}$ = 150094635296999121.
2
So $12^{27}$ 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 27 are past e, where the function is decreasing, so the larger exponent loses and $12^{27}$ > $27^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 12^{27} = 1.37370\times 10^{29}\ >\ 27^{12} = 1.50094\times 10^{17} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

12^{27}

27^{12}

$12^{27}$ is bigger: $12^{27}$ = 1.37370 × $10^{29}$ and $27^{12}$ = 1.50094 × $10^{17}$ .

Which is bigger, 122712^{27}1227 or 271227^{12}2712?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{27}$ or $27^{12}$ ?