document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=323586>Question 323586</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+log%28+2%2C+a+%29+%2B+log%28+a%2C+4%29=+3 ALIGN=MIDDLE  ALT=\"+log%28+2%2C+a+%29+%2B+log%28+a%2C+4%29=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Determine the value of 'a'<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #231528 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=231528\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Change of base formula: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C%28x%29%29=ln%28x%29%2Fln%28b%29 ALIGN=MIDDLE  ALT=\"log%28b%2C%28x%29%29=ln%28x%29%2Fln%28b%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%282%2C%28a%29%29%2Blog%28a%2C%284%29%29=3 ALIGN=MIDDLE  ALT=\"log%282%2C%28a%29%29%2Blog%28a%2C%284%29%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28a%29%2Fln%282%29%2Bln%284%29%2Fln%28a%29=3 ALIGN=MIDDLE  ALT=\"ln%28a%29%2Fln%282%29%2Bln%284%29%2Fln%28a%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the change of base formula (shown above)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28a%29%2Aln%28a%29%2Bln%284%29%2Aln%282%29=3%2Aln%282%29%2Aln%28a%29 ALIGN=MIDDLE  ALT=\"ln%28a%29%2Aln%28a%29%2Bln%284%29%2Aln%282%29=3%2Aln%282%29%2Aln%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply EVERY term by the LCD <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29%2Aln%28a%29 ALIGN=MIDDLE  ALT=\"ln%282%29%2Aln%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to clear out the fractions.\r<BR>\n" );
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document.write( "Now let's make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=ln%28a%29 ALIGN=MIDDLE  ALT=\"z=ln%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This then means that we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%2Az%2Bln%284%29%2Aln%282%29=3%2Aln%282%29%2Az ALIGN=MIDDLE  ALT=\"z%2Az%2Bln%284%29%2Aln%282%29=3%2Aln%282%29%2Az\"   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%5E2%2Bln%284%29%2Aln%282%29=3%2Aln%282%29%2Az ALIGN=MIDDLE  ALT=\"z%5E2%2Bln%284%29%2Aln%282%29=3%2Aln%282%29%2Az\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%2Az ALIGN=MIDDLE  ALT=\"z%2Az\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2-3%2Aln%282%29%2Az%2Bln%284%29%2Aln%282%29=0 ALIGN=MIDDLE  ALT=\"z%5E2-3%2Aln%282%29%2Az%2Bln%284%29%2Aln%282%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Aln%282%29%2Az ALIGN=MIDDLE  ALT=\"3%2Aln%282%29%2Az\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
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document.write( "It may be hard to see, but the equation is now in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Az%5E2%2BBz%2BC=0 ALIGN=MIDDLE  ALT=\"Az%5E2%2BBz%2BC=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. To solve for 'z', use 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-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;\" >\r<BR>\n" );
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document.write( "In this case, <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=-3%2Aln%282%29 ALIGN=MIDDLE  ALT=\"B=-3%2Aln%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=ln%284%29%2Aln%282%29 ALIGN=MIDDLE  ALT=\"C=ln%284%29%2Aln%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Plug all this into the formula to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%283%2Aln%282%29+%2B-+sqrt%28+%28-3%2Aln%282%29%29%5E2-4%281%29%28ln%284%29%2Aln%282%29%29+%29%29%2F%282%2A1%29+ ALIGN=MIDDLE   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+=+%283%2Aln%282%29+%2B-+sqrt%28+9%2A%28ln%282%29%29%5E2-4%2Aln%284%29%2Aln%282%29+%29%29%2F2+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2Aln%282%29 ALIGN=MIDDLE  ALT=\"-3%2Aln%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2A%28ln%282%29%29%5E2 ALIGN=MIDDLE  ALT=\"9%2A%28ln%282%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%289%2Aln%282%29-4%2Aln%284%29%29+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%289%2Aln%282%29-4%2Aln%284%29%29+%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29 ALIGN=MIDDLE  ALT=\"ln%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the terms under the radical.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%28ln%282%5E9%29-ln%284%5E4%29%29+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%28ln%282%5E9%29-ln%284%5E4%29%29+%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the identity  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2Aln%28x%29=ln%28x%5Ey%29 ALIGN=MIDDLE  ALT=\"y%2Aln%28x%29=ln%28x%5Ey%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=z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%28ln%28512%29-ln%28256%29%29+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%28ln%28512%29-ln%28256%29%29+%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Raise 2 to the 9th power to get 512 and raise 4 to the 4th power to get 256.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%28ln%28512%2F256%29%29+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%28ln%28512%2F256%29%29+%29%29%2F2+\"   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=ln%28A%29-ln%28B%29=ln%28A%2FB%29 ALIGN=MIDDLE  ALT=\"ln%28A%29-ln%28B%29=ln%28A%2FB%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=z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%2Aln%282%29+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%2Aln%282%29+%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%283%2Aln%282%29+%2B-+sqrt%28%28ln%282%29%29%5E2+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"z+=+%283%2Aln%282%29+%2B-+sqrt%28%28ln%282%29%29%5E2+%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29%2Aln%282%29 ALIGN=MIDDLE  ALT=\"ln%282%29%2Aln%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ln%282%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28ln%282%29%29%5E2\"   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+=+%283%2Aln%282%29+%2B-+ln%282%29%29%2F2+ ALIGN=MIDDLE  ALT=\"z+=+%283%2Aln%282%29+%2B-+ln%282%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ln%282%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28ln%282%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29 ALIGN=MIDDLE  ALT=\"ln%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The plus/minus then breaks down into two equations: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%283%2Aln%282%29+%2B+ln%282%29%29%2F2+ ALIGN=MIDDLE  ALT=\"z+=+%283%2Aln%282%29+%2B+ln%282%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%283%2Aln%282%29+-+ln%282%29%29%2F2+ ALIGN=MIDDLE  ALT=\"z+=+%283%2Aln%282%29+-+ln%282%29%29%2F2+\"   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+=+%283%2Aln%282%29+%2B+ln%282%29%29%2F2+ ALIGN=MIDDLE  ALT=\"z+=+%283%2Aln%282%29+%2B+ln%282%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the first equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%284%2Aln%282%29%29%2F2+ ALIGN=MIDDLE  ALT=\"z+=+%284%2Aln%282%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+2%2Aln%282%29+ ALIGN=MIDDLE  ALT=\"z+=+2%2Aln%282%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+ln%282%5E2%29+ ALIGN=MIDDLE  ALT=\"z+=+ln%282%5E2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the identity  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2Aln%28x%29=ln%28x%5Ey%29 ALIGN=MIDDLE  ALT=\"y%2Aln%28x%29=ln%28x%5Ey%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=z+=+ln%284%29+ ALIGN=MIDDLE  ALT=\"z+=+ln%284%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 2 to get 4.\r<BR>\n" );
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document.write( "Since we let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=ln%28a%29 ALIGN=MIDDLE  ALT=\"z=ln%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%284%29=ln%28a%29 ALIGN=MIDDLE  ALT=\"ln%284%29=ln%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=4 ALIGN=MIDDLE  ALT=\"a=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is one solution.\r<BR>\n" );
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document.write( "Now move onto the second equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%283%2Aln%282%29+-+ln%282%29%29%2F2+ ALIGN=MIDDLE  ALT=\"z+=+%283%2Aln%282%29+-+ln%282%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the second equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%282%2Aln%282%29+%29%2F2+ ALIGN=MIDDLE  ALT=\"z+=+%282%2Aln%282%29+%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+ln%282%29+ ALIGN=MIDDLE  ALT=\"z+=+ln%282%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "Again, we let  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=ln%28a%29 ALIGN=MIDDLE  ALT=\"z=ln%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, meaning that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28a%29=ln%282%29 ALIGN=MIDDLE  ALT=\"ln%28a%29=ln%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which implies that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the other solution. We know that there are only two solutions here because we're dealing with a quadratic.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the two solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=4 ALIGN=MIDDLE  ALT=\"a=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I'll leave the check for you to do. </SPAN>\n" );
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