How should I get an approximation to $(x^n - (x - 1)^n)$?

Question

I'm trying to isolate $n$ in $$\frac{x^n - (x - 1)^n}{x^n} = 0.9.$$

If I take the log in base $x$, I end up with $$\log_x(x^n - (x - 1)^n) - n = \log_x(0.9).$$

Now I don't know how to go on. Is it time to approximate this expression and move on? Should I get an approximation? Thank you.

Answer 1

$$1-(1-x^{-1})^n=0.9$$ so $$(1-x^{-1})^n=0.1,$$ $$n=\frac{\ln 0.1}{\ln(1-x^{-1})}$$ etc.

Answer 2

$$ \frac{x^n - (x - 1)^n}{x^n} = 0.9 = 1-\left(1-\frac 1x\right)^n = 0.9 $$

then

$$ \left(1-\frac 1x\right)^n = 1-0.9\rightarrow 1-\frac1x = \frac{1}{10^n} $$

and

$$ x = \frac{10^n-1}{10^n} $$

or

$$ n = \log_{10}\left(\frac{x-1}{x}\right) $$

etc