Approximate $e^{1763.02}$ (in decimal notation) using only a standard scientific calculator

Question

My attempt:

$\log(e^{1763.02})=1763.02\log e\approx 765.66$

So, $e^{1763.02}\approx 10^{765.66}$

$\implies 10^{765}<e^{1763.02}<10^{766}$

$\implies 1\cdot 10^{765}<e^{1763.02}<10\cdot 10^{765}$

$\implies e^{1763.02}=x\cdot 10^{765}$ where $1<x<10$

How do I find $x$?

Answer 1

You're just trying to evaluate $10^{765.66} = 10^{0.66} \cdot 10^{765}$. So, in your notation, $x$ is $10^{0.66} \approx 4.57$. The result is $e^{1763.02} \approx 4.57 \cdot 10^{765}$.