document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=29446>Question 29446</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following system for (x,y):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%289%2CX%29%2B+log%28Y%2C8%29=2 ALIGN=MIDDLE  ALT=\"log%289%2CX%29%2B+log%28Y%2C8%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28X%2C9%29%2B+log%288%2Cy%29=8%2F3 ALIGN=MIDDLE  ALT=\"log%28X%2C9%29%2B+log%288%2Cy%29=8%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(log to the base 9 of X) + (log to the base Y of 8) = 2<BR>\n" );
document.write( "(log to the base X of 9) + (log to the base 8 of Y) =8/3 or eight thirds </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #16245 by </B><A HREF=/tutors/aboutme.mpl?userid=sdmmadam@yahoo.com><B>sdmmadam@yahoo.com(530)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=sdmmadam@yahoo.com><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=16245\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> log9(x)+ logy(8) =2    ----(1)<BR>\n" );
document.write( "logx(9)+log8(y) = 8/3 ----(2)\r<BR>\n" );
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document.write( "Put log9(x) = a----(3) and logy(8) = b----(4)<BR>\n" );
document.write( "Putting (3) and (4) in (1), we get<BR>\n" );
document.write( " a+b = 2  ----(5)<BR>\n" );
document.write( "Using lomn(m) = 1/logm(n) in (2)<BR>\n" );
document.write( "1/log9(x) +1/logy(8) = 8/3<BR>\n" );
document.write( "That is 1/a +1/b = 8/3 ----(6) using (3) and (4)<BR>\n" );
document.write( "Multiplying by 3ab through out,<BR>\n" );
document.write( "3b+3a = 8ab<BR>\n" );
document.write( "3(b+a) = 8ab<BR>\n" );
document.write( "3X2 = 8ab ( using (5)and putting (a+b)= 2  )<BR>\n" );
document.write( "Dividing by 2<BR>\n" );
document.write( "3 = 4ab ----(7)<BR>\n" );
document.write( "We know that by formula<BR>\n" );
document.write( "(a-b)^2 = (a+b)^2 - 4ab<BR>\n" );
document.write( "(a-b)^2 = (2)^2- 3 <BR>\n" );
document.write( "(using (5) and putting a+b = 2 and using (7) and putting 4ab= 3 )<BR>\n" );
document.write( "(a-b)^2 = 4-3 = 1<BR>\n" );
document.write( "(a-b)^2  = 1<BR>\n" );
document.write( "(a-b) = 1 ----(8) (taking the positive root)<BR>\n" );
document.write( "(a+b) = 2 ----(5) (writing (5) underneath (8) to make solving look easier)<BR>\n" );
document.write( "(8) + (5) implies<BR>\n" );
document.write( "(a+a) + 0 = (1+2)<BR>\n" );
document.write( "2a=3<BR>\n" );
document.write( "a=3/2 <BR>\n" );
document.write( "a=3/2 in (5) implies b = 1/2<BR>\n" );
document.write( "Now a= 3/2 means<BR>\n" );
document.write( "log9(x) = 3/2   (using (3) )<BR>\n" );
document.write( "which implies x = (9)^(3/2)<BR>\n" );
document.write( "(we shall retain the base 9 as it is the base in the problem)<BR>\n" );
document.write( "[= (3^2)^(3/2) = (3)^3 = 27]<BR>\n" );
document.write( "(using definition logb(N) =p implies and implied by  N = (b)^p where N>0 )<BR>\n" );
document.write( "And [(m)^n](p/q) = (m)^(np/q) )<BR>\n" );
document.write( "That is x = 27<BR>\n" );
document.write( "And Now b= 1/2 means<BR>\n" );
document.write( "logy(8) = 1/2   (using (4) )<BR>\n" );
document.write( "which implies 8 = (y)^(1/2) <BR>\n" );
document.write( "(using definition logb(N) =p implies and implied by  N = (b)^p where N>0 )<BR>\n" );
document.write( "Squaring both the sides <BR>\n" );
document.write( "8^2 = y<BR>\n" );
document.write( "That is y = 8^2<BR>\n" );
document.write( "Answer: x = [9^(3/2)] (= 27) and y = (8)^(2) (=64)<BR>\n" );
document.write( "That is x = 27 and y = 64<BR>\n" );
document.write( "Verification:We used (1) for finding b using a<BR>\n" );
document.write( "Therefore we shall use (2) for verification:<BR>\n" );
document.write( "logx(9)+log8(y) = 8/3 ----(2)<BR>\n" );
document.write( "LHS = 1/log9(x) + log8(64)<BR>\n" );
document.write( "=1/[log9[9^(3/2)]] + log8 [(8)^2] <BR>\n" );
document.write( "=1/[(3/2)Xlog9(9)] + (2)Xlog8(8)   <BR>\n" );
document.write( "(since log(anything) to the base samething is =1)<BR>\n" );
document.write( "=1/[(3/2)X1]+ 2X1<BR>\n" );
document.write( "=1/[3/2]+ 2<BR>\n" );
document.write( "=2/3+2<BR>\n" );
document.write( "=(2+6)/3<BR>\n" );
document.write( "=8/3 = RHS\r<BR>\n" );
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document.write( " </SPAN>\n" );
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