How many positive integers n are there such that 10^n≤n^10?
To find the number of positive integers n that satisfy the inequality 10^n ≤ n^10, we can approach it step by step.
Step 1: Start with a small value of n and increase it gradually.
Let's begin with n = 1.
Calculating 10^1 = 10 and 1^10 = 1.
Since 10 > 1, this inequality is not satisfied.
Step 2: Increment n and check the inequality again.
Now, let's try n = 2.
Calculating 10^2 = 100 and 2^10 = 1024.
Again, 100 < 1024. So, this inequality is satisfied.
Step 3: Continue incrementing n until the inequality is no longer satisfied.
Checking with n = 3, 4, 5, and so on...
For n = 3, 10^3 = 1000 and 3^10 ≈ 59049.
So, this inequality continues to be satisfied.
For n = 4, 10^4 = 10000 and 4^10 = 1048576.
At this point, the inequality is no longer satisfied because 10000 < 1048576.
Therefore, there are only three positive integers n (2, 3, and 4) for which 10^n ≤ n^10.
To generalize, we can conclude that there are only finitely many positive integers n for which 10^n ≤ n^10.