document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=944178>Question 944178</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Simultaneous equations<BR>\n" );
document.write( "log base 4 xy = 10<BR>\n" );
document.write( "2 log base 8 x = 3 log base 8 y<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #575668 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=575668\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i'll show log base 4 as log4<BR>\n" );
document.write( "i'll show log base 8 as log8\r<BR>\n" );
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document.write( "your equations are:\r<BR>\n" );
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document.write( "log4(xy) = 10<BR>\n" );
document.write( "2log8(x) = 3log8(y)\r<BR>\n" );
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document.write( "since, in general, a*log(b) = log(b^a), your second equation can be simplified to:\r<BR>\n" );
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document.write( "log8(x^2) = log8(y^3)\r<BR>\n" );
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document.write( "this can only be true if x^2 = y^3\r<BR>\n" );
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document.write( "solve for y to get y = x^(2/3)\r<BR>\n" );
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document.write( "go back to your first equation and replace y with x^(2/3) to get:\r<BR>\n" );
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document.write( "log4(xy) = 10 becomes log4(x * x^(2/3)) = 10\r<BR>\n" );
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document.write( "since, in general, x^a * x^b equals x^(a+b), x * x^(3/3) becomes x^(5/3).\r<BR>\n" );
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document.write( "your equation becomes:\r<BR>\n" );
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document.write( "log4(x^(5/3)) = 10\r<BR>\n" );
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document.write( "this is true if and only if 4^10 = x^(5/3)\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = (4^10)^(3/5) which simplifies to:\r<BR>\n" );
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document.write( "x = 4^(30/5) which further simplifies to 4^6 which is equal to 4096.\r<BR>\n" );
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document.write( "you have x = 4096.\r<BR>\n" );
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document.write( "since y = x^(2/3), then y = 4096^(2/3) which makes y = 256.\r<BR>\n" );
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document.write( "you have x = 4096 and y = 256.\r<BR>\n" );
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document.write( "go back to your first equation of log4(xy) = 10 and replace x with 4096 and y with 256 to get:\r<BR>\n" );
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document.write( "log4(xy) = 10 becomes log4(4096*256) = 10.\r<BR>\n" );
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document.write( "this is true if and only if 4^10 = 4096*256.\r<BR>\n" );
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document.write( "simplify to get 1048576 = 1048576 so you're good.\r<BR>\n" );
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document.write( "you can also confirm that 2log8(x) = 3log(y) by replacing x with 4096 and y with 256 to get:\r<BR>\n" );
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document.write( "2log8(4096) = 3log(256)\r<BR>\n" );
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document.write( "you can use the log conversion to the base 10 formula to see if this is true.\r<BR>\n" );
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document.write( "2log8(4096) = 3log(256) becomes 2log10(4096)/log10(8) = 3log10(256)/log10(8) which you can solve using your calculator LOG function to get 8 = 8.\r<BR>\n" );
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document.write( "this confirms the values for x and y are good.\r<BR>\n" );
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document.write( "x = 4096<BR>\n" );
document.write( "y = 256\r<BR>\n" );
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