The smallest integer n for which 4 n 17 19 holds, is closest to

Question

The smallest integer n for which 4 n > 17 19 holds, is closest to

Accepted Answer

39 — Easy Take logs to base $4$. Since $17$ is just above $16=4^2$, $\log_4 17$ is just over $2$, so $n$ must just exceed $2	imes 19=38$ — the smallest integer is $39$. 1 Take $\log_4$ of both sides. $$\begin{aligned} &4^{n}>17^{19}\ &\Rightarrow\ n>19\,\log_4 17 \end{aligned}$$ 2 Estimate $\log_4 17$. Since $4^2=16 $$\begin{aligned} &\log_4 17\approx 2.043\ &\Rightarrow\ 19\,\log_4 17\approx 38.8 \end{aligned}$$ 3 Smallest integer above the bound. $$\begin{aligned} &n>38.8\ &\Rightarrow\ n_{\min

CAT 2018 Slot 2 — QA Question 22

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