<s>[INST] A simpler non-calculator proof for $17^{69}<10^{85}$ I have proved that $17^{69}<10^{85}$ by using the following inequalities:
$x<\exp\left(\dfrac{2(x-1)}{x+1}\right)$ for all $x\in \left]-1,1\right[$
and $x<{\mathrm e}^{x-1}$ for all $x\in \left] 1,+\infty \right[$, but I am looking for a simpler non-calculator proof.
My proof is the following:
\begin{align*}\frac{17^{69}}{10^{85}}&=\left(\frac{17^3}{2^3\cdot 5^4}\right)^{23}\cdot\left(\frac{5^3}{2^7}\right)^2\cdot\frac{5}{4}<\left(\frac{17^3}{2^3\cdot 5^4}\right)^{23}\cdot\frac{5}{4}=\left(\frac{4913}{5000}\right)^{23}\cdot \frac{5}{4}\\&<\left(\exp\left(\frac{2\left(\frac{4913}{5000}-1\right)}{\frac{4913}{5000}+1}\right)\right)^{23}\cdot\exp\left(\frac{5}{4}-1\right)\\&=\exp\left(-\frac{174}{431}\right)\cdot\exp\left(\frac{1}{4}\right)=\exp\left(-\frac{265}{1724}\right)<1.\end{align*}
Could anyone find a simpler non-calculator proof without using big numbers?
 [/INST] Proof that $17^{69} < 10^{85}$ without using calculator, there are many ways this can be done you'll need to understand number theory, so I'll highlight a few examples
$$17^{69} < 10^{85}$$
$$17^{69} < 10^{17×5}$$
$$17^{1/17} < 10^{5/69}$$
Remember $$\lim_{n→∞} \sqrt[n]{n} = 1$$
Meaning the value of 17^{1/17} is between $1$ to $1.4$
$$17^{1/17} < 10^{5/69}$$
Using long division $\frac{5}{69} = 0.07246$
$$17^{1/17} < 10^{x}$$
But $10^x$ can also be between $1$ to $1.4$ if $x$ is from $0$ to $≈0.1$
So the the value $\frac{5}{69}$ is too small, therefore
$$17^{69} \mathbb{<} 10^{85}$$
Another way
$$17^{69} < 10^{85}$$
$$17^{69} < 10^{17×5}$$
$$17^{69/17} < 10^{5}$$
Using long division $\frac{69}{17} = 4.0588$
$$17^{4.0588} < 10^{5}$$
If we assume that $17^{4.0588} = 20^x$, since the base became bigger, the power would be smaller for them to be equal
$17 → 20$$, $4.0588 → ≈3$
$$17^{4.0588} ≈ 20^3$$
$$≈20^3 < 10^5$$
But we know clearly that $8000 < 10000$
Therefore 
$$17^{69} \mathbb{<} 10^{85}$$
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