Which is bigger, $10^{29}$ or $29^{10}$ ?

Q: Which is bigger, 10^29 or 29^10?

10^29 is bigger: 10^29 = 1.00000 × 10^29 and 29^10 = 4.20707 × 10^14.

$10^{29}$ is bigger: $10^{29}$ = 1.00000 × $10^{29}$ and $29^{10}$ = 4.20707 × $10^{14}$ .

10^{29} = 1.00000\times 10^{29}\ >\ 29^{10} = 4.20707\times 10^{14}

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The curve ln x / x, which decides every comparison of a^b with b^a. 10^29 against 29^10 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: $10^{29}$ = 100000000000000000000000000000 and $29^{10}$ = 420707233300201.
2
So $10^{29}$ 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 10 and 29 are past e, where the function is decreasing, so the larger exponent loses and $10^{29}$ > $29^{10}$ .
$\frac{\ln 10}{10}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 10^{29} = 1.00000\times 10^{29}\ >\ 29^{10} = 4.20707\times 10^{14} \]
\[ \frac{\ln 10}{10}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

10^{29}

29^{10}

$10^{29}$ is bigger: $10^{29}$ = 1.00000 × $10^{29}$ and $29^{10}$ = 4.20707 × $10^{14}$ .

Which is bigger, 102910^{29}1029 or 291029^{10}2910?

How to work it out

Questions

Similar comparisons

Which is bigger, $10^{29}$ or $29^{10}$ ?