document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194878>Question 1194878</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this question:log2{√(x³-2x²+x)}=1+log2(x-1) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #827157 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=827157\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C%28sqrt%28x%5E3-2x%5E2%2Bx%29%29%29=1%2Blog%282%2C%28x-1%29%29 ALIGN=MIDDLE  ALT=\"log%282%2C%28sqrt%28x%5E3-2x%5E2%2Bx%29%29%29=1%2Blog%282%2C%28x-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "There are, as nearly always, many different paths to solving this equation.  My solution below is only one option.<br><BR>\n" );
document.write( "(1) I would first express the constant \"1\" in terms of log base 2.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C%28sqrt%28x%5E3-2x%5E2%2Bx%29%29%29=log%282%2C%282%29%29%2Blog%282%2C%28x-1%29%29 ALIGN=MIDDLE  ALT=\"log%282%2C%28sqrt%28x%5E3-2x%5E2%2Bx%29%29%29=log%282%2C%282%29%29%2Blog%282%2C%28x-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(2) On the right, use the rule that the sum of logs is the log of the product.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C%28sqrt%28x%5E3-2x%5E2%2Bx%29%29%29=log%282%2C%282%28x-1%29%29%29 ALIGN=MIDDLE  ALT=\"log%282%2C%28sqrt%28x%5E3-2x%5E2%2Bx%29%29%29=log%282%2C%282%28x-1%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(3) The logs of the two expressions are equal, so the expressions are equal.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E3-2x%5E2%2Bx%29=2%28x-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E3-2x%5E2%2Bx%29=2%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(4) Square both sides to get rid of the radical.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2x%5E2%2Bx=4%28x-1%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E3-2x%5E2%2Bx=4%28x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Instead of expanding on the right, simplify on the left....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%5E2-2x%2B1%29=4%28x-1%29%5E2 ALIGN=MIDDLE  ALT=\"x%28x%5E2-2x%2B1%29=4%28x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x-1%29%5E2=4%28x-1%29%5E2 ALIGN=MIDDLE  ALT=\"x%28x-1%29%5E2=4%28x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x-1%29%5E2-4%28x-1%29%5E2=0 ALIGN=MIDDLE  ALT=\"x%28x-1%29%5E2-4%28x-1%29%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%28x-1%29%5E2=0 ALIGN=MIDDLE  ALT=\"%28x-4%29%28x-1%29%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%28x%2B1%29%28x-1%29=0 ALIGN=MIDDLE  ALT=\"%28x-4%29%28x%2B1%29%28x-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The potential solutions are x=4, x=-1, and x=1.<br><BR>\n" );
document.write( "(5) We need to check for extraneous solutions, because (1) at one point in the solution process we squared both sides of the equation and (2) log(A) is defined only if A is positive.<br><BR>\n" );
document.write( "The expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C%28x-1%29%29 ALIGN=MIDDLE  ALT=\"log%282%2C%28x-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the original equation makes the solutions x=-1 and x=1 invalid; we need to see if x=4 satisfies the original equation.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C%28sqrt%284%5E3-2%284%5E2%29%2B2%29%29%29=log%282%2Csqrt%28%2864-32%2B4%29%29%29=log%282%2C%28sqrt%2836%29%29%29=log%282%2C%286%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Blog%282%2C%284-1%29%29=log%282%2C%282%284%29-2%29%29=log%282%2C%286%29%29 ALIGN=MIDDLE  ALT=\"1%2Blog%282%2C%284-1%29%29=log%282%2C%282%284%29-2%29%29=log%282%2C%286%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "x=4 satisfies the original equation, so it is the only solution.<br><BR>\n" );
document.write( "ANSWER: x=4<br><BR>\n" );
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