document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=991818>Question 991818</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1/log2a+1/log4a+1/log8a+......upto n term  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #611556 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52903)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=611556\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Flog%282%2Ca%29 ALIGN=MIDDLE  ALT=\"1%2Flog%282%2Ca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Flog%284%2Ca%29 ALIGN=MIDDLE  ALT=\"1%2Flog%284%2Ca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Flog%288%2Ca%29 ALIGN=MIDDLE  ALT=\"1%2Flog%288%2Ca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + . . . . . upto n term<BR>\n" );
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document.write( "Notice that \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Flog%28b%2Ca%29 ALIGN=MIDDLE  ALT=\"1%2Flog%28b%2Ca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2Cb%29 ALIGN=MIDDLE  ALT=\"log%28a%2Cb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "It is the consequence of the &nbsp;\"base change formula\" &nbsp;for logarithms &nbsp;(see any &nbsp;<B>Algebra</B>&nbsp; textbook or the lesson &nbsp;<A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/logarithm/change-of-base-formula-for-logarithms.lesson>Change of Base Formula for logarithms</A>&nbsp; in this site). \r<BR>\n" );
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document.write( "Therefore, &nbsp;we can re-write the formula in the form\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Flog%282%2Ca%29 ALIGN=MIDDLE  ALT=\"1%2Flog%282%2Ca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Flog%284%2Ca%29 ALIGN=MIDDLE  ALT=\"1%2Flog%284%2Ca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Flog%288%2Ca%29 ALIGN=MIDDLE  ALT=\"1%2Flog%288%2Ca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + . . . + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Flog%282%5En%2Ca%29 ALIGN=MIDDLE  ALT=\"1%2Flog%282%5En%2Ca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C2%29 ALIGN=MIDDLE  ALT=\"log%28a%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C4%29 ALIGN=MIDDLE  ALT=\"log%28a%2C4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C8%29 ALIGN=MIDDLE  ALT=\"log%28a%2C8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + . . . + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C2%5En%29 ALIGN=MIDDLE  ALT=\"log%28a%2C2%5En%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Next, &nbsp;notice that &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C4%29 ALIGN=MIDDLE  ALT=\"log%28a%2C4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Alog%28a%2C2%29 ALIGN=MIDDLE  ALT=\"2%2Alog%28a%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C8%29 ALIGN=MIDDLE  ALT=\"log%28a%2C8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Alog%28a%2C2%29 ALIGN=MIDDLE  ALT=\"3%2Alog%28a%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, . . . , &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C2%5En%29 ALIGN=MIDDLE  ALT=\"log%28a%2C2%5En%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2Alog%28a%2C2%29 ALIGN=MIDDLE  ALT=\"n%2Alog%28a%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Thus you have, &nbsp;actually, &nbsp;the sum  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C2%29 ALIGN=MIDDLE  ALT=\"log%28a%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Alog%28a%2C2%29 ALIGN=MIDDLE  ALT=\"2%2Alog%28a%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Alog%28a%2C2%29 ALIGN=MIDDLE  ALT=\"3%2Alog%28a%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + . . . + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2Alog%28a%2C2%29 ALIGN=MIDDLE  ALT=\"n%2Alog%28a%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C2%29 ALIGN=MIDDLE  ALT=\"log%28a%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.(1 + 2 + 3 + . . . +n),\r<BR>\n" );
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document.write( "which is equal to &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C2%29 ALIGN=MIDDLE  ALT=\"log%28a%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28n%28n%2B1%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"%28%28n%28n%2B1%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<U>Answer</U>. &nbsp;The sum is equal to &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C2%29 ALIGN=MIDDLE  ALT=\"log%28a%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28n%28n%2B1%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"%28%28n%28n%2B1%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Flog%282%2Ca%29%29 ALIGN=MIDDLE  ALT=\"%281%2Flog%282%2Ca%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28n%28n%2B1%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"%28%28n%28n%2B1%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Thank you for submitting so exciting problem.\r<BR>\n" );
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