Which is bigger, $4^{26}$ or $26^{4}$ ?

Q: Which is bigger, 4^26 or 26^4?

4^26 is bigger: 4^26 = 4.50359 × 10^15 and 26^4 = 456976.

$4^{26}$ is bigger: $4^{26}$ = 4.50359 × $10^{15}$ and $26^{4}$ = 456976.

4^{26} = 4.50359\times 10^{15}\ >\ 26^{4} = 456976

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^26 against 26^4 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: $4^{26}$ = 4503599627370496 and $26^{4}$ = 456976.
2
So $4^{26}$ 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 4 and 26 are past e, where the function is decreasing, so the larger exponent loses and $4^{26}$ > $26^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 4^{26} = 4.50359\times 10^{15}\ >\ 26^{4} = 456976 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

4^{26}

26^{4}

$4^{26}$ is bigger: $4^{26}$ = 4.50359 × $10^{15}$ and $26^{4}$ = 456976.

Which is bigger, 4264^{26}426 or 26426^{4}264?

How to work it out

Questions

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Which is bigger, $4^{26}$ or $26^{4}$ ?