Find least positive integer $n$ which satisfies the condition

Question

Find least positive integer $n$ which satisfies the condition

• $| {0.5}^{\frac{1}{n}}-1|<10^{-3} $

• $0.9^{n}<10^{-3}$

What I have tried is I have split the first inequality took log on both side .. but I came to know it is getting complicated

Answer 1

for the second one we get by taking the logarithm $$n\ln\left(\frac{9}{10}\right)<\ln(10^{-3}$$ therefore $$n>\frac{10^{-3}}{\ln(0.9)}$$ we get $$n\geq 66$$ for the second one we get $$n>\frac{\ln(0.5)}{\ln(1-10^{-3})}$$ therefore $$n\geq 693$$

Answer 2

\begin{array}{c} | 0.5^{\frac 1n}-1| < \frac{1}{1000}\\ 1-\frac{1}{1000} < 0.5^{\frac 1n} < 1+\frac{1}{1000} \\ \frac{999}{1000} < 0.5^{\frac 1n} < \frac{1001}{1000} \\ \log 999 - 3 < -\frac 1n \log \dfrac 12 < \log 1001 - 3 \\ \log 999 - 3 < \frac 1n \log 2 < \log 1001 - 3 \\ \dfrac{\log 999 - 3}{\log 2} < \frac 1n < \dfrac{\log 1001 - 3}{\log 2} \\ \end{array}

$\dfrac{\log 999 - 3}{\log 2} < 0 < \frac 1n$ is always true for positive integer values of $n$.

\begin{align} \frac 1n < \dfrac{\log 1001 - 3}{\log 2} \\ n > \dfrac{\log 2}{\log 1001 - 3} \\ n > 693.49\dots \\ \end{align}

The least integer value of $n$ is $n = 694$