Question 1106296
1)
{{{5^3=125}}} <--> {{{log(5,125)=3.000}}}
{{{system(124<125,log(5,125)=3)}}} --> {{{log(5,124)<3}}}
 
{{{5^2=25}}} <--> {{{log(5,25)=2}}}
{{{system(24<25,log(5,25)=2)}}} --> {{{log(5,24)<2}}}
 
So, the numbers 25, 26, ....124 have logarithm with characteristic 2.
Those are the first 124 minus the first 24:
{{{124-24=highlight(100)}}} positive integers.
 
2)=
{{{log(10,5^25)=25log(10,5)=approx}}}{{{25*0.69897=17.47425}}}
As {{{17<17.47425<18}}} , {{{10^17<5^25<10^18}}} ,
so {{{5^25}}} has {{{highlight(18)}}} digits, just like {{{10^17}}} ,
which is a 1 followed by 17 zeros.
 
We can even calculate that
{{{5^25=10^25/2^25=10*10^24/(2^10*2^10*2^5)}}}{{{"="}}}{{{10*10^24/(1024*1024*32)}}}
{{{"="}}}{{{10*10^24/(1048576*32)}}}{{{"="}}}{{{10*10^24/33554432=10*10^24/(3.3554432*10^7)}}}
{{{"= about"}}}{{{2.98*10^((24-7))=about}}}{{{2.98*10^17}}}