Which is bigger, $6^{22}$ or $22^{6}$ ?

Q: Which is bigger, 6^22 or 22^6?

6^22 is bigger: 6^22 = 1.31621 × 10^17 and 22^6 = 113379904.

$6^{22}$ is bigger: $6^{22}$ = 1.31621 × $10^{17}$ and $22^{6}$ = 113379904.

6^{22} = 1.31621\times 10^{17}\ >\ 22^{6} = 113379904

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. 6^22 against 22^6 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: $6^{22}$ = 131621703842267136 and $22^{6}$ = 113379904.
2
So $6^{22}$ 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 6 and 22 are past e, where the function is decreasing, so the larger exponent loses and $6^{22}$ > $22^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 6^{22} = 1.31621\times 10^{17}\ >\ 22^{6} = 113379904 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

6^{22}

22^{6}

$6^{22}$ is bigger: $6^{22}$ = 1.31621 × $10^{17}$ and $22^{6}$ = 113379904.

Which is bigger, 6226^{22}622 or 22622^{6}226?

How to work it out

Questions

Similar comparisons

Which is bigger, $6^{22}$ or $22^{6}$ ?