document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=881537>Question 881537</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that log5 xy = 2 log25 x + 2 log25 y. Hence or otherwise, find the values of x and y which satisfy the equations log5 xy = 10 and (log25 y/log25 x) 3/2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #532291 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=532291\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you want to prove:\r<BR>\n" );
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document.write( "Log5(xy) = 2Log25(x) + 2Log25(y)\r<BR>\n" );
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document.write( "2Log25(x) is equal to Log25(x^2)<BR>\n" );
document.write( "2Log25(y) is equal to Log25(y^2)\r<BR>\n" );
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document.write( "Log25(x^2) + Log25(y^2) is equal to Log25(x^2 * y^2) which is equal to Log25((xy)^2).\r<BR>\n" );
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document.write( "Your original equation becomes:\r<BR>\n" );
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document.write( "Log5(xy) = Log25((xy)^2)\r<BR>\n" );
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document.write( "set Log5(xy) equal to a.<BR>\n" );
document.write( "set Log25((xy)^2) equal to b.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "Log5(xy) = a<BR>\n" );
document.write( "Log25((xy)^2) = b\r<BR>\n" );
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document.write( "If we can prove that a = b, then that proves that Log5(xy) = Log25((xy)^2) which proves the original equation is true.\r<BR>\n" );
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document.write( "The basic definition of logs states that:\r<BR>\n" );
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document.write( "Log5(xy) = a if and only if 5^a = xy\r<BR>\n" );
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document.write( "Likewise, the basic definition of logs states that:\r<BR>\n" );
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document.write( "Log25((xy)^2) = b if and only if 25^b = (xy)^2.\r<BR>\n" );
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document.write( "25^b is the same as (5^2)^b which makes this last equation equivalent to:\r<BR>\n" );
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document.write( "(5^2)^b = (xy)^2 which can also be expressed as:\r<BR>\n" );
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document.write( "5^(2b) = (xy)^2 which can also be expressed as:\r<BR>\n" );
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document.write( "(5^b)^2 = (xy)^2.\r<BR>\n" );
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document.write( "(5^b)^2 can only be equal to (xy)^2 if 5^b = xy.\r<BR>\n" );
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document.write( "you can prove this by just taking the square root of both sides of the equation and you will get 5^b = xy.\r<BR>\n" );
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document.write( "we now have:\r<BR>\n" );
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document.write( "5^a = xy<BR>\n" );
document.write( "5^b = xy\r<BR>\n" );
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document.write( "This can only be true if a = b.\r<BR>\n" );
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document.write( "Since we have proven that a = b, we have also proven that Log5(xy) is equal to Log25((xy)^2) which proves the original equation is true.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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