document.write( "Question 1106296: 1) how many positive integers have characteristic 2 when base is 5.
\n" ); document.write( "2) find the number of digits in
\n" ); document.write( " 5^25 \n" ); document.write( "

Algebra.Com's Answer #721258 by KMST(5328) $\"\"$ $\"About$

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1)
\n" ); document.write( " $\"5%5E3=125\"$ <--> $\"log%285%2C125%29=3.000\"$
\n" ); document.write( " $\"system%28124%3C125%2Clog%285%2C125%29=3%29\"$ --> $\"log%285%2C124%29%3C3\"$
\n" ); document.write( "
\n" ); document.write( " $\"5%5E2=25\"$ <--> $\"log%285%2C25%29=2\"$
\n" ); document.write( " $\"system%2824%3C25%2Clog%285%2C25%29=2%29\"$ --> $\"log%285%2C24%29%3C2\"$
\n" ); document.write( "
\n" ); document.write( "So, the numbers 25, 26, ....124 have logarithm with characteristic 2.
\n" ); document.write( "Those are the first 124 minus the first 24:
\n" ); document.write( " $\"124-24=highlight%28100%29\"$ positive integers.
\n" ); document.write( "
\n" ); document.write( "2)=
\n" ); document.write( " $\"log%2810%2C5%5E25%29=25log%2810%2C5%29=approx\"$ $\"25%2A0.69897=17.47425\"$
\n" ); document.write( "As $\"17%3C17.47425%3C18\"$ , $\"10%5E17%3C5%5E25%3C10%5E18\"$ ,
\n" ); document.write( "so $\"5%5E25\"$ has $\"highlight%2818%29\"$ digits, just like $\"10%5E17\"$ ,
\n" ); document.write( "which is a 1 followed by 17 zeros.
\n" ); document.write( "
\n" ); document.write( "We can even calculate that
\n" ); document.write( " $\"5%5E25=10%5E25%2F2%5E25=10%2A10%5E24%2F%282%5E10%2A2%5E10%2A2%5E5%29\"$ $\"%22=%22\"$ $\"10%2A10%5E24%2F%281024%2A1024%2A32%29\"$
\n" ); document.write( " $\"%22=%22\"$ $\"10%2A10%5E24%2F%281048576%2A32%29\"$ $\"%22=%22\"$ $\"10%2A10%5E24%2F33554432=10%2A10%5E24%2F%283.3554432%2A10%5E7%29\"$
\n" ); document.write( " $\"%22=+about%22\"$ $\"2.98%2A10%5E%28%2824-7%29%29=about\"$ $\"2.98%2A10%5E17\"$ \n" ); document.write( "

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