Which is bigger, $15^{22}$ or $22^{15}$ ?

Q: Which is bigger, 15^22 or 22^15?

15^22 is bigger: 15^22 = 7.48182 × 10^25 and 22^15 = 1.36880 × 10^20.

$15^{22}$ is bigger: $15^{22}$ = 7.48182 × $10^{25}$ and $22^{15}$ = 1.36880 × $10^{20}$ .

15^{22} = 7.48182\times 10^{25}\ >\ 22^{15} = 1.36880\times 10^{20}

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The curve ln x / x, which decides every comparison of a^b with b^a. 15^22 against 22^15 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: $15^{22}$ = 74818276426792144775390625 and $22^{15}$ = 136880068015412051968.
2
So $15^{22}$ 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 15 and 22 are past e, where the function is decreasing, so the larger exponent loses and $15^{22}$ > $22^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 15^{22} = 7.48182\times 10^{25}\ >\ 22^{15} = 1.36880\times 10^{20} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

15^{22}

22^{15}

$15^{22}$ is bigger: $15^{22}$ = 7.48182 × $10^{25}$ and $22^{15}$ = 1.36880 × $10^{20}$ .

Which is bigger, 152215^{22}1522 or 221522^{15}2215?

How to work it out

Questions

Similar comparisons

Which is bigger, $15^{22}$ or $22^{15}$ ?