Animated Solution for Mathematics - Basic Mathematics: If $n$ is a natural number such that $n=p_1^{{\alpha }_1}\cdot p_2^{{\alpha }_2}\cdot p_3^{{\alpha }_3}\dots p_k^{{\alpha }_k}$ and $p_1,p_2,\dots p_k$ are distinct primes, then show that $\ln n\ge k\ln 2$.

Imagine you are standing before the vast, infinite landscape of natural numbers. Every single one of them, from the smallest prime to the most gargantuan integer, shares a secret identity. They are all built from the same fundamental bricks: prime numbers.

This is the Fundamental Theorem of Arithmetic. When we write $n=p_1^{{\alpha }_1}\cdot p_2^{{\alpha }_2}\dots p_k^{{\alpha }_k}$, we are essentially looking at the DNA of the number $n$. Today, we are going to explore how this structure dictates the behavior of the natural logarithm of $n$.

We are tasked with proving that $\ln n\ge k\ln 2$. At first glance, this might seem like a leap. How does a product of prime powers relate to a simple sum involving $k$ and $\ln 2$?

The answer lies in the power of logarithms. Logarithms are the great translators of mathematics; they turn the complexity of multiplication into the simplicity of addition. When we apply the natural logarithm to our prime factorization, we get:

Using the product rule, $\ln (ab)=\ln a+\ln b$, we can break this massive product into a sum:

And with the power rule, $\ln (a^b)=b\ln a$, those exponents that were hiding in the shadows jump to the front, giving us a clean, manageable expression:

Now, let us look at the individual components of this sum. We know that $p_i$ are distinct prime numbers. What is the smallest prime number? It is $2$.

Therefore, for any prime $p_i$, we must have $p_i\ge 2$. Because the natural logarithm function $f(x)=\ln x$ is strictly increasing, applying it to both sides of this inequality preserves the direction: $\ln p_i\ge \ln 2$.

Next, consider the exponents ${\alpha }_i$. Since $n$ is a natural number, its prime factors must have positive integer exponents. Thus, ${\alpha }_i\ge 1$.

Now, we have two simple inequalities: ${\alpha }_i\ge 1$ and $\ln p_i\ge \ln 2$. Since all these values are positive, we can multiply them together without fear of flipping the inequality sign:

We have reached the final stage of our journey. We know that each term in our summation, ${\alpha }_i\ln p_i$, is at least $\ln 2$. If we sum these terms from $i=1$ to $k$, we are essentially adding $\ln 2$ to itself $k$ times:

On the left, we have our original $\ln n$. On the right, we have $k$ copies of $\ln 2$, which is simply $k\ln 2$. Thus, we arrive at the elegant conclusion:

This result is more than just an inequality; it is a testament to the structure of numbers. The number of distinct prime factors $k$ acts as a fundamental lower bound for the logarithm of $n$. It is a beautiful, simple, and powerful truth that lies at the heart of number theory.