document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=970739>Question 970739</A>: <I><SPAN STYLE=\"word-break: break-all;\"> prove that if a series of numbers are in GP,their logarithms are in AP </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #593375 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=593375\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> geometric progression is An = A1 * r^(n-1)\r<BR>\n" );
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document.write( "arithmetric progression is An = A1 + (n-1)*d\r<BR>\n" );
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document.write( "start with the geometric progression of An = A1 * r^(n-1)\r<BR>\n" );
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document.write( "take the log of both sides of that equation to get:\r<BR>\n" );
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document.write( "log(An) = log(A1 * r^(n-1))\r<BR>\n" );
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document.write( "since log(a*b) = log(a) + log(b), this equation becomes:\r<BR>\n" );
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document.write( "log(An) = log(A1) + log(r^(n-1))\r<BR>\n" );
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document.write( "since log(a^b) = b*log(a), this equation becomes:\r<BR>\n" );
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document.write( "log(An) = log(A1) + (n-1)*log(r)\r<BR>\n" );
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document.write( "that's an arithmetic rogression with the common difference equal to log(r).\r<BR>\n" );
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document.write( "we'll take an example:\r<BR>\n" );
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document.write( "A1 = 100<BR>\n" );
document.write( "r = 1.5<BR>\n" );
document.write( "n = 5\r<BR>\n" );
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document.write( "geometric progression:\r<BR>\n" );
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document.write( "An = A1 * (r^(n-1) which becomes:<BR>\n" );
document.write( "A5 = 100 * 1.5^4 which becomes:<BR>\n" );
document.write( "A5 = 506.25\r<BR>\n" );
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document.write( "now we want to find log(A5).\r<BR>\n" );
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document.write( "if we are correct, then log(A5) should be equal to log(506.25).\r<BR>\n" );
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document.write( "start with:\r<BR>\n" );
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document.write( "log(A1) = log(100)<BR>\n" );
document.write( "r = 1.5<BR>\n" );
document.write( "n = 5\r<BR>\n" );
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document.write( "use the arithmetic progression of:\r<BR>\n" );
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document.write( "log(A5) = log(A1) + (n-1)*log(r) which becomes:<BR>\n" );
document.write( "log(A5) = log(100) + 4*log(1.5) which becomes:<BR>\n" );
document.write( "log(A5) = log(100) + log(1.5^4) whcih becomes:<BR>\n" );
document.write( "log(A5) = log(100*1.5^4) which becomes:<BR>\n" );
document.write( "log(A5) = log(506.25)\r<BR>\n" );
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document.write( "A5 = 506.25<BR>\n" );
document.write( "log(A5) = log(506.25)\r<BR>\n" );
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document.write( "looks like we're good.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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