Which is bigger, $18^{25}$ or $25^{18}$ ?

Q: Which is bigger, 18^25 or 25^18?

18^25 is bigger: 18^25 = 2.40886 × 10^31 and 25^18 = 1.45519 × 10^25.

$18^{25}$ is bigger: $18^{25}$ = 2.40886 × $10^{31}$ and $25^{18}$ = 1.45519 × $10^{25}$ .

18^{25} = 2.40886\times 10^{31}\ >\ 25^{18} = 1.45519\times 10^{25}

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The curve ln x / x, which decides every comparison of a^b with b^a. 18^25 against 25^18 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: $18^{25}$ = 24088659210943104643915283693568 and $25^{18}$ = 14551915228366851806640625.
2
So $18^{25}$ 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 18 and 25 are past e, where the function is decreasing, so the larger exponent loses and $18^{25}$ > $25^{18}$ .
$\frac{\ln 18}{18}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 18^{25} = 2.40886\times 10^{31}\ >\ 25^{18} = 1.45519\times 10^{25} \]
\[ \frac{\ln 18}{18}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

18^{25}

25^{18}

$18^{25}$ is bigger: $18^{25}$ = 2.40886 × $10^{31}$ and $25^{18}$ = 1.45519 × $10^{25}$ .

Which is bigger, 182518^{25}1825 or 251825^{18}2518?

How to work it out

Questions

Similar comparisons

Which is bigger, $18^{25}$ or $25^{18}$ ?