IIT-JEE 1984Question 1
If $n$ is a natural number such that $n=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}\cdots p_k^{\alpha_k}$ and $p_1,p_2,\ldots,p_k$ are distinct primes, then show that $\ln n\ge k\ln2$.
Sequence and Series · Miscellaneous
Answerproved
Solution
Formulas used. (1) $\ln$ is increasing: $u\ge v>0\Rightarrow\ln u\ge\ln v$Step 1 - Lower bound for $n$. Every prime $p_i\ge2$ and every exponent $\alpha_i\ge1$, so $p_i^{\alpha_i}\ge2$: $$n=p_1^{\alpha_1}\cdots p_k^{\alpha_k}\ge2^k .$$ Step 2 - Take logs. $$\ln n\ge\ln2^k=k\ln2 .$$ Answer: proved