Which is bigger, $25^{26}$ or $26^{25}$ ?

Q: Which is bigger, 25^26 or 26^25?

25^26 is bigger: 25^26 = 2.22044 × 10^36 and 26^25 = 2.36773 × 10^35.

$25^{26}$ is bigger: $25^{26}$ = 2.22044 × $10^{36}$ and $26^{25}$ = 2.36773 × $10^{35}$ .

25^{26} = 2.22044\times 10^{36}\ >\ 26^{25} = 2.36773\times 10^{35}

Explain it step by step in chat Make a short video of it

The curve ln x / x, which decides every comparison of a^b with b^a. 25^26 against 26^25 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: $25^{26}$ = 2220446049250313080847263336181640625 and $26^{25}$ = 236773830007967588876795164938469376.
2
So $25^{26}$ 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 26 are past e, where the function is decreasing, so the larger exponent loses and $25^{26}$ > $26^{25}$ .
$\frac{\ln 25}{25}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 25^{26} = 2.22044\times 10^{36}\ >\ 26^{25} = 2.36773\times 10^{35} \]
\[ \frac{\ln 25}{25}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

25^{26}

26^{25}

$25^{26}$ is bigger: $25^{26}$ = 2.22044 × $10^{36}$ and $26^{25}$ = 2.36773 × $10^{35}$ .

Which is bigger, 252625^{26}2526 or 262526^{25}2625?

How to work it out

Questions

Similar comparisons

Which is bigger, $25^{26}$ or $26^{25}$ ?