Which is greater? $2^{128}$ or $3^{81}$

Which is greater?

$$2^{128}$$ or $$3^{81}$$

No calculators allowed.

Solution [Click here to expand/collapse]

$2^{128} = 340282366920938463463374607431768211456$

$3^{81} = 443426488243037769948249630619149892803$

So $3^{81} \gt 2^{128}$. QED (in about 4 hours).

Just kidding :)

Here is a proof without calculators.

Raise both sides to power of $\frac{1}{16}$.
We get $$2^8 \text{ vs } 3^{\frac{1}{16}}. 3^5$$ Now $3^{1/16} \gt e^{1/16} \gt 1 + \frac{1}{16}$ (using $e^x \gt 1 + x$)

Since $13 \times 16 = (15-2)(15+1) \lt 15^2 = 225 \lt 243$

We have that $1 + \frac{1}{16} \gt 1 + \frac{13}{243} = \frac{2^8}{3^5}$.
QED.