Which is bigger, $15^{23}$ or $23^{15}$ ?

Q: Which is bigger, 15^23 or 23^15?

15^23 is bigger: 15^23 = 1.12227 × 10^27 and 23^15 = 2.66635 × 10^20.

$15^{23}$ is bigger: $15^{23}$ = 1.12227 × $10^{27}$ and $23^{15}$ = 2.66635 × $10^{20}$ .

15^{23} = 1.12227\times 10^{27}\ >\ 23^{15} = 2.66635\times 10^{20}

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. 15^23 against 23^15 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: $15^{23}$ = 1122274146401882171630859375 and $23^{15}$ = 266635235464391245607.
2
So $15^{23}$ 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 23 are past e, where the function is decreasing, so the larger exponent loses and $15^{23}$ > $23^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 15^{23} = 1.12227\times 10^{27}\ >\ 23^{15} = 2.66635\times 10^{20} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

15^{23}

23^{15}

$15^{23}$ is bigger: $15^{23}$ = 1.12227 × $10^{27}$ and $23^{15}$ = 2.66635 × $10^{20}$ .

Which is bigger, 152315^{23}1523 or 231523^{15}2315?

How to work it out

Questions

Similar comparisons

Which is bigger, $15^{23}$ or $23^{15}$ ?