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, you can start exploring the values of n.
Let's start with n = 1.
10^1 = 10
1^10 = 1
Since 10 > 1, the inequality is not satisfied for n = 1.
Next, let's try n = 2.
10^2 = 100
2^10 = 1024
Again, 100 < 1024, so the inequality is not satisfied for n = 2.
We continue this process and examine a few more values:
n = 3:
10^3 = 1000
3^10 = 59049
1000 < 59049
n = 4:
10^4 = 10000
4^10 = 1048576
10000 < 1048576
n = 5:
10^5 = 100000
5^10 = 9765625
100000 < 9765625
As you can see, for every positive integer n greater than 5, 10^n is always smaller than n^10. Therefore, there are no positive integers that satisfy the inequality 10^n ≤ n^10.
To arrive at this conclusion, you need to examine the values of n and compare the magnitude of 10^n with n^10.