document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210603>Question 1210603</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I came across this in an old book:\r<BR>\n" );
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document.write( "Determine logarithm of 6253.\r<BR>\n" );
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document.write( "In table, the mantissa is .7959 (The same mantissa of 6250).   The next larger is .7966, mantissa of 6260.\r<BR>\n" );
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document.write( "6260 - 6250 = 10 (difference of 10).\r<BR>\n" );
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document.write( ".7966 - .7959 = 0.0007\r<BR>\n" );
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document.write( "This is part I don't understand:\r<BR>\n" );
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document.write( "Each increase of 1 unit (6250 increase of 10 to 6260), mantissas of their logs increase by 0.0007, an increase of .1 of 0.0007. 3 added to 0250 will add .3 of 0.0007 or 0.0002 to the mantissa of 6253.  \r<BR>\n" );
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document.write( "Why is .1 and .3 added ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854279 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5350)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854279\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A table with entries like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C2%2Cnumber%2Cmantisa%2C6250%2C7959%2C6360%2C7966%29 ALIGN=MIDDLE  ALT=\"matrix%283%2C2%2Cnumber%2Cmantisa%2C6250%2C7959%2C6360%2C7966%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "can be used to find the mantissa (sort-of the decimal part) of a logarithm in base 10 of a number with 4 significant digits.<BR>\n" );
document.write( "The table does not have a mantissa for 6253, so you have to interpolate.<BR>\n" );
document.write( "Not worrying for now about decimal places, we are looking for a mantissa between 7959 and 7966.<BR>\n" );
document.write( "For an increase between 6250 and 6260 of 10, there is an increase of 7=7966-7959<BR>\n" );
document.write( "for the mantissa.<BR>\n" );
document.write( "That is a 0.7 increase in mantissa for each unit increase in the number.<BR>\n" );
document.write( "The increase in mantissa corresponding the difference between 6250 and 6253 is<BR>\n" );
document.write( "3x0.7=2.1, but we round it to 2.<BR>\n" );
document.write( "So the mantissa for 6253 is 7959+2=7961<BR>\n" );
document.write( "Not worrying for now about decimal places, we are looking for a mantissa between 7959 and 7966.<BR>\n" );
document.write( "The mantissa is the same, for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%286250%29 ALIGN=MIDDLE  ALT=\"log%286250%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%286.250%29=0.7959 ALIGN=MIDDLE  ALT=\"log%286.250%29=0.7959\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%280.006250%29 ALIGN=MIDDLE  ALT=\"log%280.006250%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . Those logarithms <BR>\n" );
document.write( "No matter where the decimal point is located, those logarithms are all found by adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.7959 ALIGN=MIDDLE  ALT=\"0.7959\"   DISABLED_style= \"max-width:90%; height:auto;\" > plus some integer.<BR>\n" );
document.write( "The integer depends on how the number of digits to the left of the decimal point<BR>\n" );
document.write( "6250 and 6250.00 have 4 digits to the left, just like 1000=10^3<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%281000%29=log%2810%5E3%29=3 ALIGN=MIDDLE  ALT=\"log%281000%29=log%2810%5E3%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810000%29=log%2810%5E4%29=4 ALIGN=MIDDLE  ALT=\"log%2810000%29=log%2810%5E4%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "6253 is more than 1000 and less than 10000, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%286253%29 ALIGN=MIDDLE  ALT=\"log%286253%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is more than 3 but less than 4.<BR>\n" );
document.write( "To find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%286253%29 ALIGN=MIDDLE  ALT=\"log%286253%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the mantissa <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.7961 ALIGN=MIDDLE  ALT=\"0.7961\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%286253%29=3%2B0.7961 ALIGN=MIDDLE  ALT=\"log%286253%29=3%2B0.7961\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28log%286253=3.761%29%29 ALIGN=MIDDLE  ALT=\"highlight%28log%286253=3.761%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "What about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%280.6253%29 ALIGN=MIDDLE  ALT=\"log%280.6253%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( "0.6253 is between 1 and 0.1, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%281%29=log%2810%5E0%29=0 ALIGN=MIDDLE  ALT=\"log%281%29=log%2810%5E0%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%280.1%29=log%281%2F10%29=log%2810%5E-1%29=-1 ALIGN=MIDDLE  ALT=\"log%280.1%29=log%281%2F10%29=log%2810%5E-1%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%280.6253%29 ALIGN=MIDDLE  ALT=\"log%280.6253%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be more than -1 and less than 0.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%280.6253%29=-1%2B0.7961 ALIGN=MIDDLE  ALT=\"log%280.6253%29=-1%2B0.7961\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%280.6253%29=-0.2039 ALIGN=MIDDLE  ALT=\"log%280.6253%29=-0.2039\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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