SOLUTION: 1) how many positive integers have characteristic 2 when base is 5. 2) find the number of digits in 5^25

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: 1) how many positive integers have characteristic 2 when base is 5. 2) find the number of digits in 5^25      Log On


   

Question 1106296: 1) how many positive integers have characteristic 2 when base is 5.
2) find the number of digits in
5^25
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
1)
5%5E3=125 <--> log%285%2C125%29=3.000
system%28124%3C125%2Clog%285%2C125%29=3%29 --> log%285%2C124%29%3C3

5%5E2=25 <--> log%285%2C25%29=2
system%2824%3C25%2Clog%285%2C25%29=2%29 --> log%285%2C24%29%3C2

So, the numbers 25, 26, ....124 have logarithm with characteristic 2.
Those are the first 124 minus the first 24:
124-24=highlight%28100%29 positive integers.

2)=
log%2810%2C5%5E25%29=25log%2810%2C5%29=approx25%2A0.69897=17.47425
As 17%3C17.47425%3C18 , 10%5E17%3C5%5E25%3C10%5E18 ,
so 5%5E25 has highlight%2818%29 digits, just like 10%5E17 ,
which is a 1 followed by 17 zeros.

We can even calculate that
5%5E25=10%5E25%2F2%5E25=10%2A10%5E24%2F%282%5E10%2A2%5E10%2A2%5E5%29%22=%2210%2A10%5E24%2F%281024%2A1024%2A32%29
%22=%2210%2A10%5E24%2F%281048576%2A32%29%22=%2210%2A10%5E24%2F33554432=10%2A10%5E24%2F%283.3554432%2A10%5E7%29
%22=+about%222.98%2A10%5E%28%2824-7%29%29=about2.98%2A10%5E17