document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=886288>Question 886288</A>: <I><SPAN STYLE=\"word-break: break-all;\"> log_2((x^2)+ 1)&#8722;log_4(x^2) = 1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #535882 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=535882\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i get x = 1\r<BR>\n" );
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document.write( "start with log2(x^2 + 1) - log4(x^2) = 1\r<BR>\n" );
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document.write( "take log4(x^2) and set it equal to y<BR>\n" );
document.write( "you get log4(x^2) = y if and only if 4^y = x^2<BR>\n" );
document.write( "4^y is equivalent to 2^(2y) and you get:<BR>\n" );
document.write( "2^(2y) = x^2<BR>\n" );
document.write( "you get 2^(2y) = x^2 if and only if log2(x^2) = 2y\r<BR>\n" );
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document.write( "solve for y and you get:<BR>\n" );
document.write( "y = log2^(x^2) / 2<BR>\n" );
document.write( "since log4^(x^2) is also equal to y, you get:<BR>\n" );
document.write( "log2^(x^2) / 2 = log4(x^2)\r<BR>\n" );
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document.write( "replace log4(x^2) with log2(x^2)/2 in your original equation and you get:<BR>\n" );
document.write( "log2(x^2 + 1) - log2(x^2)/2 = 1\r<BR>\n" );
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document.write( "multiply both sides of this equation by 2 to get:<BR>\n" );
document.write( "2*log2(x^2 + 1) - log2(x^2) = 2\r<BR>\n" );
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document.write( "since 2*log2(x^2 + 1) is equivalent to log2((x^2+1)^2), then your equation becomes log2((x^2+1)^2) - log2(x^2) = 2\r<BR>\n" );
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document.write( "since log(a) - log(b) = log(a/b), your equation becomes:<BR>\n" );
document.write( "log2(((x^2+1)^2)/x^2) = 2\r<BR>\n" );
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document.write( "this is true if and only if 2^2 = ((x^2+1)^2/x^2)\r<BR>\n" );
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document.write( "simplify this and you will get:\r<BR>\n" );
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document.write( "4 = (x^4 + 2x^2 + 1) / x^2\r<BR>\n" );
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document.write( "multiply both sides of this equation by x^2 and you get:\r<BR>\n" );
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document.write( "4x^2 = x^4 + 2x^2 + 1\r<BR>\n" );
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document.write( "subtract 4x^2 from both sides of this equation and you get:\r<BR>\n" );
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document.write( "x^4 - 2x^2 + 1 = 0\r<BR>\n" );
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document.write( "let y = x^2 and the equation becomes:\r<BR>\n" );
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document.write( "y^2 - 2y + 1 = 0\r<BR>\n" );
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document.write( "factor this equation and you get:\r<BR>\n" );
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document.write( "(y-1)^2 = 0\r<BR>\n" );
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document.write( "solve for y and you get y = 1\r<BR>\n" );
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document.write( "since y = x^2, then you get x^2 = 1\r<BR>\n" );
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document.write( "solve for x and you get x = +/- sqrt(1) which becomes x = +/- 1\r<BR>\n" );
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document.write( "these should be your solutions.\r<BR>\n" );
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document.write( "plug them into your original equation to see if they're good.\r<BR>\n" );
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document.write( "your original equation is:\r<BR>\n" );
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document.write( "log2(x^2 + 1) - log4(x^2) = 1\r<BR>\n" );
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document.write( "log2(1^2 + 1) = log2(2) = 1<BR>\n" );
document.write( "log2((-1)^2 + 1) = log2(2) = 1\r<BR>\n" );
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document.write( "log4(1^2) = log4(1) = 0<BR>\n" );
document.write( "log4((-1)^2) = log4(1) = 0\r<BR>\n" );
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document.write( "your original equation of log2(x^2 + 1) - log4(x^2) = 1 becomes 1 - 0 = 1 which is true.\r<BR>\n" );
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document.write( "your solutions are x = 1 and x = -1\r<BR>\n" );
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document.write( " </SPAN>\n" );
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