document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1134866>Question 1134866</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a geometric progression series find the value of r if logr+logr^2+logr^4+logr^8+...+logr^32=63 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #752361 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=752361\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> it is unfortunate that they used the designation of r in the log because that gets it confused with the rate in the formula of Sn = A1 * (1 - r^n) / (1 - r).\r<BR>\n" );
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document.write( "that formula represents the sum of a geometric series.\r<BR>\n" );
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document.write( "i changed the r to an x and the problem now reads.\r<BR>\n" );
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document.write( "log(x) + log(x^2) + log(x^4) + log(x^8) + ... log(x^32) = 63.\r<BR>\n" );
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document.write( "the missing term between log(x^8) and log(x^32) looks like it needs to be log(x^16).\r<BR>\n" );
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document.write( "the series then becomes log(x) + log(x^2) + log(x^4) + log(x^8) + log(x^16) + log(x^32) = 63.\r<BR>\n" );
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document.write( "since log(x^n) = n * log(x), that series can now be shown as:\r<BR>\n" );
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document.write( "log(x) + 2 * log(x) + 4 * log(x) + 8 * log(x) + 16 * log(x) + 32 * log(x) = 63\r<BR>\n" );
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document.write( "the common ratio is 2, because each term in the series doubles in size.\r<BR>\n" );
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document.write( "there are 6 terms in the series.\r<BR>\n" );
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document.write( "the first terms is log(x).\r<BR>\n" );
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document.write( "the formula for the sum of a geometric series is Sn = A1 (1 - r^n) / (1 - r)\r<BR>\n" );
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document.write( "when A1 = log(x) and n = 6 and r = 2, this formula becomes:\r<BR>\n" );
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document.write( "Sn = log(x) * (1 - 2^6) / (1 - 2).\r<BR>\n" );
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document.write( "since Sn = 63, this formula becomes 63 = log(x) * (1 - 2^6) / (1 - 2)\r<BR>\n" );
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document.write( "simplify this formula to get 63 = log(x) * -63 / -1.\r<BR>\n" );
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document.write( "simplify to get 63 = log(x) * 63.\r<BR>\n" );
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document.write( "solve for log(x) to get log(x) = 1\r<BR>\n" );
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document.write( "this is true if and only if 10^1 = x.\r<BR>\n" );
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document.write( "this makes x = 10.\r<BR>\n" );
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document.write( "when x = 10 and A1 = log(x) and r = 2 and n = 6 and Sn = 63, Sn = A1 * (1 - r^n) / (1 - r) becomes 63 = log(10) * (1 - 2^6) / (1 - 2) which becomes 63 = 1 * -63 / -1 which becomes 63 = 63 which is true.\r<BR>\n" );
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document.write( "your solution appears to be be x = 10.\r<BR>\n" );
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document.write( "going back to the original series where r was used in place of x, you would get:\r<BR>\n" );
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document.write( "log(10) + log(10^2) + log(10^4) + log(10^8) + log(1-^16) + log(10^32) = 63\r<BR>\n" );
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document.write( "you can use your calculator to confirm that this last equation is true.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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