document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1077085>Question 1077085</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve 3log_(5)y-log_(y)5=2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #691651 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=691651\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> this winds up being able to solve a quadratic equation.<BR>\n" );
document.write( "the math is horrendous, but i used an online quadratic equation calculator to solve to the degree of accuracy required.\r<BR>\n" );
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document.write( "the concept is as follows:\r<BR>\n" );
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document.write( "start with 3log5(y) - logy(5) = 2\r<BR>\n" );
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document.write( "conver to base of 10 log as follows:\r<BR>\n" );
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document.write( "log5(y) = log(6)/log(5)\r<BR>\n" );
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document.write( "logy(5) = log(5)/log(y)\r<BR>\n" );
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document.write( "equation becomes 3log(y)/log(5) - log(5)/log(y) = 2\r<BR>\n" );
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document.write( "log(anything) is assumeed to be log10(anything).\r<BR>\n" );
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document.write( "your calculator can handle log10(anything) by use of the calculator LOG function.\r<BR>\n" );
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document.write( "so your equation has become:\r<BR>\n" );
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document.write( "3log(y)/log(5) - log(5)/log(y) = 2\r<BR>\n" );
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document.write( "multiply both sides of this equation by log(5)log(y) to get:\r<BR>\n" );
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document.write( "3log(y)^2 - log(5)^2 = 2log(5)log(y)\r<BR>\n" );
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document.write( "subtract 2log(5)log(y) from both sides of the equation to get:\r<BR>\n" );
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document.write( "3log(y)^2 - 2log(5)log(y) - log(5)^2 = 0\r<BR>\n" );
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document.write( "let x = log(y).\r<BR>\n" );
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document.write( "equation becomes 3x^2 - 2log(5)x - log(5)^2 = 0\r<BR>\n" );
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document.write( "this is a quadratic equation that can be solved by using the quadratic formula.\r<BR>\n" );
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document.write( "i spared myself the drudgery by using the following online quadratic equation solver:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.mathsisfun.com/quadratic-equation-solver.html\" target = \"_blank\">https://www.mathsisfun.com/quadratic-equation-solver.html</a>\r<BR>\n" );
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document.write( "since this solver couldn't handle -2log(5) or -log(5)^2, i had to convert them to fractions.\r<BR>\n" );
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document.write( "-2log(5) became -1.397940009\r<BR>\n" );
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document.write( "-log(5)^2 became -.48859067\r<BR>\n" );
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document.write( "these were input into the quadratic solve and the result was:\r<BR>\n" );
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document.write( "x = 0.69897000443178 or x = -0.23299000143178\r<BR>\n" );
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document.write( "since x is equal to log(y), then the solution became:\r<BR>\n" );
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document.write( "log(y) = 0.69897000443178 or log(y) = -0.23299000143178\r<BR>\n" );
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document.write( "log(y) = 0.69897000443178 if and only if 10^0.69897000443178 = y\r<BR>\n" );
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document.write( "this led to y = 5.000000001 or y = .5848035477\r<BR>\n" );
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document.write( "it remained to place these values into the original equation to see if that equation holds true.\r<BR>\n" );
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document.write( "it holds true for y = 5.000000001\r<BR>\n" );
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document.write( "it also holds true for y = .5848035477\r<BR>\n" );
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document.write( "the solution from the quadratic solver is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/041802.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "i also solved the equation graphically as shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/041801.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the graphical solutions are rounded to 3 decimal places but they confirm the alebraic solutuion is correct.\r<BR>\n" );
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document.write( "in the graph, i used x instead of y.\r<BR>\n" );
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document.write( "same equation with a change in variable name.\r<BR>\n" );
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document.write( "the intersection of the 2 graphs is the solution, with the correct value of the original y being the x-coordinate of the coordinate pair.\r<BR>\n" );
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document.write( "for example, the coordinate pair of (.585,2) says that the value of x is .585.\r<BR>\n" );
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document.write( "you would simply read that as y = .585.\r<BR>\n" );
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document.write( "similarly, the coordinate pair of (5,2) says that the value of x is 5.\r<BR>\n" );
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document.write( "you would simply read that as y = 5.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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