Which is bigger, $15^{26}$ or $26^{15}$ ?

Q: Which is bigger, 15^26 or 26^15?

15^26 is bigger: 15^26 = 3.78767 × 10^30 and 26^15 = 1.67725 × 10^21.

$15^{26}$ is bigger: $15^{26}$ = 3.78767 × $10^{30}$ and $26^{15}$ = 1.67725 × $10^{21}$ .

15^{26} = 3.78767\times 10^{30}\ >\ 26^{15} = 1.67725\times 10^{21}

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. 15^26 against 26^15 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: $15^{26}$ = 3787675244106352329254150390625 and $26^{15}$ = 1677259342285725925376.
2
So $15^{26}$ 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 15 and 26 are past e, where the function is decreasing, so the larger exponent loses and $15^{26}$ > $26^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 26}{26}$

LaTeX source of the formulas on this page

\[ 15^{26} = 3.78767\times 10^{30}\ >\ 26^{15} = 1.67725\times 10^{21} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 26}{26} \]

Questions

Which is bigger,

15^{26}

26^{15}

$15^{26}$ is bigger: $15^{26}$ = 3.78767 × $10^{30}$ and $26^{15}$ = 1.67725 × $10^{21}$ .

Which is bigger, 152615^{26}1526 or 261526^{15}2615?

How to work it out

Questions

Similar comparisons

Which is bigger, $15^{26}$ or $26^{15}$ ?