Which is bigger, $25^{29}$ or $29^{25}$ ?

Q: Which is bigger, 25^29 or 29^25?

25^29 is bigger: 25^29 = 3.46944 × 10^40 and 29^25 = 3.63036 × 10^36.

$25^{29}$ is bigger: $25^{29}$ = 3.46944 × $10^{40}$ and $29^{25}$ = 3.63036 × $10^{36}$ .

25^{29} = 3.46944\times 10^{40}\ >\ 29^{25} = 3.63036\times 10^{36}

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The curve ln x / x, which decides every comparison of a^b with b^a. 25^29 against 29^25 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: $25^{29}$ = 34694469519536141888238489627838134765625 and $29^{25}$ = 3630362123627258663193028251474330749.
2
So $25^{29}$ 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 25 and 29 are past e, where the function is decreasing, so the larger exponent loses and $25^{29}$ > $29^{25}$ .
$\frac{\ln 25}{25}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 25^{29} = 3.46944\times 10^{40}\ >\ 29^{25} = 3.63036\times 10^{36} \]
\[ \frac{\ln 25}{25}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

25^{29}

29^{25}

$25^{29}$ is bigger: $25^{29}$ = 3.46944 × $10^{40}$ and $29^{25}$ = 3.63036 × $10^{36}$ .

Which is bigger, 252925^{29}2529 or 292529^{25}2925?

How to work it out

Questions

Similar comparisons

Which is bigger, $25^{29}$ or $29^{25}$ ?