Question

number of positive integers which have the characteristic 4 when the base of log is 5

Answer 1

2500

Answer 2

The characteristic of $\log_b N$ is the integer part of $\log_b N$. If the characteristic is $k$, it means $k \le \log_b N < k+1$.

In this problem, the base is 5 and the characteristic is 4. So, we have:
$4 \le \log_5 N < 5$

To find the range of $N$, we exponentiate the inequality with base 5:
$5^4 \le 5^{\log_5 N} < 5^5$

Since $N$ is a positive integer, $5^{\log_5 N} = N$. So, the inequality becomes:
$5^4 \le N < 5^5$

Calculate the values of $5^4$ and $5^5$:
$5^4 = 5 \times 5 \times 5 \times 5 = 625$
$5^5 = 5 \times 5 \times 5 \times 5 \times 5 = 3125$

The inequality is therefore:
$625 \le N < 3125$

We are looking for the number of positive integers $N$ that satisfy this condition. The integers $N$ start from 625 and go up to 3124.
The set of integers is $\{625, 626, 627, \dots, 3124\}$.

To find the number of integers in the range $[a, b)$, which is $a \le N < b$, the number of integers is $b - a$.
In this case, $a = 625$ and $b = 3125$.
The number of integers is $3125 - 625$.

Number of integers = $3125 - 625 = 2500$.

Thus, there are 2500 positive integers that have the characteristic 4 when the base of the logarithm is 5.