Which is bigger, $14^{29}$ or $29^{14}$ ?

Q: Which is bigger, 14^29 or 29^14?

14^29 is bigger: 14^29 = 1.72867 × 10^33 and 29^14 = 2.97558 × 10^20.

$14^{29}$ is bigger: $14^{29}$ = 1.72867 × $10^{33}$ and $29^{14}$ = 2.97558 × $10^{20}$ .

14^{29} = 1.72867\times 10^{33}\ >\ 29^{14} = 2.97558\times 10^{20}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^29 against 29^14 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: $14^{29}$ = 1728673739677471101567216945987584 and $29^{14}$ = 297558232675799463481.
2
So $14^{29}$ 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 14 and 29 are past e, where the function is decreasing, so the larger exponent loses and $14^{29}$ > $29^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 14^{29} = 1.72867\times 10^{33}\ >\ 29^{14} = 2.97558\times 10^{20} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

14^{29}

29^{14}

$14^{29}$ is bigger: $14^{29}$ = 1.72867 × $10^{33}$ and $29^{14}$ = 2.97558 × $10^{20}$ .

Which is bigger, 142914^{29}1429 or 291429^{14}2914?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{29}$ or $29^{14}$ ?