Which is bigger, $8^{26}$ or $26^{8}$ ?

Q: Which is bigger, 8^26 or 26^8?

8^26 is bigger: 8^26 = 3.02231 × 10^23 and 26^8 = 208827064576.

$8^{26}$ is bigger: $8^{26}$ = 3.02231 × $10^{23}$ and $26^{8}$ = 208827064576.

8^{26} = 3.02231\times 10^{23}\ >\ 26^{8} = 208827064576

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The curve ln x / x, which decides every comparison of a^b with b^a. 8^26 against 26^8 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: $8^{26}$ = 302231454903657293676544 and $26^{8}$ = 208827064576.
2
So $8^{26}$ 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 8 and 26 are past e, where the function is decreasing, so the larger exponent loses and $8^{26}$ > $26^{8}$ .
$\frac{\ln 8}{8}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 8^{26} = 3.02231\times 10^{23}\ >\ 26^{8} = 208827064576 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

8^{26}

26^{8}

$8^{26}$ is bigger: $8^{26}$ = 3.02231 × $10^{23}$ and $26^{8}$ = 208827064576.

Which is bigger, 8268^{26}826 or 26826^{8}268?

How to work it out

Questions

Similar comparisons

Which is bigger, $8^{26}$ or $26^{8}$ ?