Explain how does the second step follows from the first?

Question

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Explain how does the second step follows from the first?

Answer 1

We have $\frac{1}{2^n}\frac{1}{2}<\frac{1}{2}\frac{1}{10^6}$, hence

$2^n>10^6$ and therefore $n \log_{10}2 > \log_{10}10^6=6$

Answer 2

We have $$\frac{1}{2^{n+1}} < 0.5 \times 10^{-6}$$ $$\Rightarrow \frac{1}{2^n}\frac{1}{2} < \frac{1}{2}\times 10^{-6}$$ $$\Rightarrow \frac{1}{2^n} < 10^{-6}$$ Now, using the concept of logarithms, we have, $$\log(\frac{1}{2^n}) < \log (10^{-6})$$ $$\Rightarrow -n\log 2 < -6$$ When there is a negative sign on both sides of the inequality, the inequality reverses. Thus, $$n > \frac{6}{\log 2} \sim 19.9$$ Hope it helps.