Which is bigger, $4^{29}$ or $29^{4}$ ?

Q: Which is bigger, 4^29 or 29^4?

4^29 is bigger: 4^29 = 2.88230 × 10^17 and 29^4 = 707281.

$4^{29}$ is bigger: $4^{29}$ = 2.88230 × $10^{17}$ and $29^{4}$ = 707281.

4^{29} = 2.88230\times 10^{17}\ >\ 29^{4} = 707281

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^29 against 29^4 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: $4^{29}$ = 288230376151711744 and $29^{4}$ = 707281.
2
So $4^{29}$ 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 4 and 29 are past e, where the function is decreasing, so the larger exponent loses and $4^{29}$ > $29^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 4^{29} = 2.88230\times 10^{17}\ >\ 29^{4} = 707281 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

4^{29}

29^{4}

$4^{29}$ is bigger: $4^{29}$ = 2.88230 × $10^{17}$ and $29^{4}$ = 707281.

Which is bigger, 4294^{29}429 or 29429^{4}294?

How to work it out

Questions

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Which is bigger, $4^{29}$ or $29^{4}$ ?