Which is bigger, $6^{26}$ or $26^{6}$ ?

Q: Which is bigger, 6^26 or 26^6?

6^26 is bigger: 6^26 = 1.70581 × 10^20 and 26^6 = 308915776.

$6^{26}$ is bigger: $6^{26}$ = 1.70581 × $10^{20}$ and $26^{6}$ = 308915776.

6^{26} = 1.70581\times 10^{20}\ >\ 26^{6} = 308915776

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. 6^26 against 26^6 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: $6^{26}$ = 170581728179578208256 and $26^{6}$ = 308915776.
2
So $6^{26}$ 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 6 and 26 are past e, where the function is decreasing, so the larger exponent loses and $6^{26}$ > $26^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 6^{26} = 1.70581\times 10^{20}\ >\ 26^{6} = 308915776 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

6^{26}

26^{6}

$6^{26}$ is bigger: $6^{26}$ = 1.70581 × $10^{20}$ and $26^{6}$ = 308915776.

Which is bigger, 6266^{26}626 or 26626^{6}266?

How to work it out

Questions

Similar comparisons

Which is bigger, $6^{26}$ or $26^{6}$ ?