document.write( "Question 1194801: How many digits are in the expansion of 5^30? How do you do it without a calculator. Show your steps. \n" ); document.write( "

Algebra.Com's Answer #827070 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "How many digits are in the expansion of 5^30?
\n" ); document.write( "How do you do it without a calculator. Show your steps.
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\n" ); document.write( "\n" ); document.write( "             The idea of the solution is to estimate logarithm base 10 of the number N = $\"5%5E30\"$ .\r
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\n" ); document.write( "\n" ); document.write( "             Then use the fact that if log(N) is concluded between two consecutive integer numbers n and (n+1), \r
\n" ); document.write( "\n" ); document.write( "            n <= log(N) < (n+1),   then the integer number N has (n+1) digit.\r
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document.write( "Without using a calculator, you find from the logarithmic tables\r\n" );
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document.write( "    log(5) = 0.69897  (approximately)\r\n" );
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document.write( "(logarithms base 10; for the logarithms table see many Internet sources, for example, this one\r\n" );
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document.write( "http://www.sosmath.com/tables/logtable/logtable.html)\r\n" );
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document.write( "Then   = 30*log(5) = 30*0.69897 = 20.969...\r\n" );
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document.write( "It means that the number   has 21 digits in base 10 numerical system.\r\n" );
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