document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=174880>Question 174880</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve the logarithm\r<BR>\n" );
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document.write( "2log3z-log3(z-2)=2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #129950 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=129950\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Alog%283%2C%28z%29%29-log%283%2C%28z-2%29%29=2 ALIGN=MIDDLE  ALT=\"2%2Alog%283%2C%28z%29%29-log%283%2C%28z-2%29%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation. \r<BR>\n" );
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document.write( "Take note that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%3E0 ALIGN=MIDDLE  ALT=\"z%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for the first log) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z-2%3E0 ALIGN=MIDDLE  ALT=\"z-2%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%3E2 ALIGN=MIDDLE  ALT=\"z%3E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for the second log). \r<BR>\n" );
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document.write( "So the domain is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%3E2 ALIGN=MIDDLE  ALT=\"z%3E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (this interval works for BOTH logs)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%283%2C%28z%5E2%29%29-log%283%2C%28z-2%29%29=2 ALIGN=MIDDLE  ALT=\"log%283%2C%28z%5E2%29%29-log%283%2C%28z-2%29%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite the first log using the identity  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2Alog%28b%2C%28x%29%29=log%28b%2C%28x%5Ey%29%29 ALIGN=MIDDLE  ALT=\"y%2Alog%28b%2C%28x%29%29=log%28b%2C%28x%5Ey%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%283%2C%28%28z%5E2%29%2F%28z-2%29%29%29=2 ALIGN=MIDDLE  ALT=\"log%283%2C%28%28z%5E2%29%2F%28z-2%29%29%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the logs using the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C%28A%29%29-log%28b%2C%28B%29%29=log%28b%2C%28A%2FB%29%29 ALIGN=MIDDLE  ALT=\"log%28b%2C%28A%29%29-log%28b%2C%28B%29%29=log%28b%2C%28A%2FB%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E2=%28z%5E2%29%2F%28z-2%29 ALIGN=MIDDLE  ALT=\"3%5E2=%28z%5E2%29%2F%28z-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite the equation using the property: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C%28x%29%29=y ALIGN=MIDDLE  ALT=\"log%28b%2C%28x%29%29=y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ====> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Ey=x ALIGN=MIDDLE  ALT=\"b%5Ey=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=%28z%5E2%29%2F%28z-2%29 ALIGN=MIDDLE  ALT=\"9=%28z%5E2%29%2F%28z-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 3 to get 9\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%28z-2%29=z%5E2 ALIGN=MIDDLE  ALT=\"9%28z-2%29=z%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z-2 ALIGN=MIDDLE  ALT=\"z-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9z-18=z%5E2 ALIGN=MIDDLE  ALT=\"9z-18=z%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-z%5E2%2B9z-18=0 ALIGN=MIDDLE  ALT=\"-z%5E2%2B9z-18=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2 ALIGN=MIDDLE  ALT=\"z%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
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document.write( "Notice we have a quadratic equation in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=az%5E2%2Bbz%2Bc ALIGN=MIDDLE  ALT=\"az%5E2%2Bbz%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-1 ALIGN=MIDDLE  ALT=\"a=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=9 ALIGN=MIDDLE  ALT=\"b=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-18 ALIGN=MIDDLE  ALT=\"c=-18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for z\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"z+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-%289%29+%2B-+sqrt%28+%289%29%5E2-4%28-1%29%28-18%29+%29%29%2F%282%28-1%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28-%289%29+%2B-+sqrt%28+%289%29%5E2-4%28-1%29%28-18%29+%29%29%2F%282%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-1 ALIGN=MIDDLE  ALT=\"a=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=9 ALIGN=MIDDLE  ALT=\"b=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-18 ALIGN=MIDDLE  ALT=\"c=-18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-9+%2B-+sqrt%28+81-4%28-1%29%28-18%29+%29%29%2F%282%28-1%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28-9+%2B-+sqrt%28+81-4%28-1%29%28-18%29+%29%29%2F%282%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81 ALIGN=MIDDLE  ALT=\"81\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-9+%2B-+sqrt%28+81-72+%29%29%2F%282%28-1%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28-9+%2B-+sqrt%28+81-72+%29%29%2F%282%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28-1%29%28-18%29 ALIGN=MIDDLE  ALT=\"4%28-1%29%28-18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72 ALIGN=MIDDLE  ALT=\"72\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-9+%2B-+sqrt%28+9+%29%29%2F%282%28-1%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28-9+%2B-+sqrt%28+9+%29%29%2F%282%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72 ALIGN=MIDDLE  ALT=\"72\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81 ALIGN=MIDDLE  ALT=\"81\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-9+%2B-+sqrt%28+9+%29%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"z+=+%28-9+%2B-+sqrt%28+9+%29%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-9+%2B-+3%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"z+=+%28-9+%2B-+3%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-9+%2B+3%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"z+=+%28-9+%2B+3%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-9+-+3%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"z+=+%28-9+-+3%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the expression. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-6%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"z+=+%28-6%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=++%28-12%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"z+=++%28-12%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+3 ALIGN=MIDDLE  ALT=\"z+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+6 ALIGN=MIDDLE  ALT=\"z+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify. \r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+3 ALIGN=MIDDLE  ALT=\"z+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+6 ALIGN=MIDDLE  ALT=\"z+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Take note that both of these solutions are within the domain <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%3E2 ALIGN=MIDDLE  ALT=\"z%3E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So they are valid solutions.<BR>\n" );
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