document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=970605>Question 970605</A>: <I><SPAN STYLE=\"word-break: break-all;\"> x=log a base 2a, y=log 2a base 3a, z=log 3a base 4a then prove that xyz + 1=2yz </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #593214 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=593214\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you are given:\r<BR>\n" );
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document.write( "x = L2a(a)<BR>\n" );
document.write( "y = L3a(2a)<BR>\n" );
document.write( "z = L4a(3a)\r<BR>\n" );
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document.write( "you are asked to prove that xyz+1 = 2yz\r<BR>\n" );
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document.write( "note that Lb(x) means log of x to the base of b.<BR>\n" );
document.write( "for example: L2a(a) means log(a) to the base of 2a.\r<BR>\n" );
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document.write( "you can convert everything to base of 10 using the log base conversion formula of:\r<BR>\n" );
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document.write( "Lb(x) = L(x)/L(b)\r<BR>\n" );
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document.write( "that means log of x to the base of b is equivalent to log of x to the base of 10 divided by log of b to the base of 10.\r<BR>\n" );
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document.write( "your original equations get converted as follows:\r<BR>\n" );
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document.write( "x = L2a(a) becomes x = L(a)/L(2a)<BR>\n" );
document.write( "y = L3a(2a) becomes y = L(2a)/L(3a)<BR>\n" );
document.write( "z = L4a(3a) becomes z = L(3a)/L(4a)\r<BR>\n" );
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document.write( "applying these equivalencies, you get:\r<BR>\n" );
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document.write( "2yz = 2*L(2a)/L(3a)*L(3a)/L4a) which becomes:\r<BR>\n" );
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document.write( "2yz = 2*L(2a)/L(4a).\r<BR>\n" );
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document.write( "since 2*L(2a) becomes L((2a)^2) which becomes L(4a^2), then:\r<BR>\n" );
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document.write( "2yz = 2*L(2a)/L(4a) bcomes:\r<BR>\n" );
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document.write( "2yz = L(4a^2)/L(4a)\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "xyz+1 = L(a)/L(2a)*L(2a)/L(3a)*L(3a)/L(4a)+1 which becomes:\r<BR>\n" );
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document.write( "xyz+1 = L(a)/L(4a)+1 which becomes:\r<BR>\n" );
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document.write( "xyz+1 = L(a)/L(4a)+L(4a)/L(4a) which becomes:\r<BR>\n" );
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document.write( "xyz+1 = (L(a)+L(4a))/L(4a)\r<BR>\n" );
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document.write( "since L(a)+L(4a) is equal to L(a*4a) which is equal to L(4a^2), then:\r<BR>\n" );
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document.write( "xyz+1 = (L(a)+L(4a))/L(4a) becomes:\r<BR>\n" );
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document.write( "xyz+1 = L(4a^2)/L(4a).\r<BR>\n" );
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document.write( "since 2yz = L(4a^2)/L(4a) and xyz+1 = L(4a^2)/L(4a), then:\r<BR>\n" );
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document.write( "2yz = xyz+1\r<BR>\n" );
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document.write( "end of proof\r<BR>\n" );
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document.write( " </SPAN>\n" );
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