Question 1194801
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How many digits are in the expansion of 5^30? 
How do you do it without a calculator. Show your steps.
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                The idea of the solution is to estimate logarithm base  10  of the number  N = {{{5^30}}}.


                Then use the fact that if   log(N)  is concluded between two consecutive integer numbers  n  and  (n+1), 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;n <= log(N) < (n+1), &nbsp;&nbsp;then the integer number &nbsp;N  &nbsp;has &nbsp;(n+1) &nbsp;digit.



<pre>
Without using a calculator, you find from the logarithmic tables

    log(5) = 0.69897  (approximately)


(logarithms base 10; for the logarithms table see many Internet sources, for example, this one

http://www.sosmath.com/tables/logtable/logtable.html)



Then  {{{log((5^30))}}} = 30*log(5) = 30*0.69897 = 20.969...


It means that the number {{{5^30}}}  has 21 digits in base 10 numerical system.
</pre>

Solved and explained.