I want to express below number in $ e^x $ format,
$$ (1.05)^{1000} $$
so I used following Identity
$$ e^x = \left( 1+ \frac{1}{x} \right)^x $$
$$ = \left( \left(1 + \frac{1}{20} \right)^{20}\right)^{50} $$
$$ = e^{50} $$
is it correct or something I am doing wrong. please help me.
Where did you get this identity from? The one that I know exists, looks like this:
$ lim_{x \rightarrow 0} (1 + \frac{1}{x})^x = e$
You want to find $x$ such that $e^x = (1.05)^{1000}$. We can take natural log on both sides and solve it like this,
$ \Rightarrow ln(e^x) = ln((1.05)^{1000}) \\ \Rightarrow x \cdot ln(e) = 1000 ln(1.05) \\ \Rightarrow x = 1000 \times 0.04879$
where $ln(1.05) = 0.04879$.