document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=719670>Question 719670</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am having a difficult time solving a few problems: \r<BR>\n" );
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document.write( "here is the first problem: ln(x-3)+ln(3x+1)=10<BR>\n" );
document.write( "I am not sure where to start the process to solve this problem, any help or guidance is greatly appreciated.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #441451 by </B><A HREF=/tutors/aboutme.mpl?userid=swincher4391><B>swincher4391(1107)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=swincher4391><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=swincher4391><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=441451\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The key to this problem is to understand the logarithmic properties. The property invoked in this problem is the multiplication property. That is ln(a) + ln(b) = ln(ab).\r<BR>\n" );
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document.write( "Using this we know that ln(x-3)+ln(3x+1) is really  ln[(x-3)(3x+1)].\r<BR>\n" );
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document.write( "So ln[(x-3)(3x+1)] = 10.\r<BR>\n" );
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document.write( "By FOIL:\r<BR>\n" );
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document.write( "ln[3x^2 -8x -3] = 10\r<BR>\n" );
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document.write( "Since ln is the natural log [with base e], we can EXPONENTIATE both sides by a base of e.\r<BR>\n" );
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document.write( "e^ln[3x^2-8x-3] = e^10\r<BR>\n" );
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document.write( "3x^2 - 8x - 3 = e^10\r<BR>\n" );
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document.write( "3x^2 - 8x - 3-e^10 = 0\r<BR>\n" );
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document.write( "Use the quadratic formula to clean this up:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288+%2B-+sqrt%2864+-+%284%2A3%2A%28-3-e%5E10%29%29%29%29%2F+6%29 ALIGN=MIDDLE  ALT=\"%288+%2B-+sqrt%2864+-+%284%2A3%2A%28-3-e%5E10%29%29%29%29%2F+6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Which is roughly: \r<BR>\n" );
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document.write( "x= 87.036, -84.369\r<BR>\n" );
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document.write( "Remember that domain of the natural log is (0, infinity). So -84.369 will give us a negative in the logarithm which is not allowed. So the only acceptable solution is 87.036 or in exact form:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288+%2B+sqrt%28100+%2B+12e%5E10%29%29%29%29%2F+6%29 ALIGN=MIDDLE  ALT=\"%288+%2B+sqrt%28100+%2B+12e%5E10%29%29%29%29%2F+6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which may be simplified to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288+%2B+sqrt%284%2A%2825%2B3e%5E10%29%29%29%2F6 ALIGN=MIDDLE  ALT=\"%288+%2B+sqrt%284%2A%2825%2B3e%5E10%29%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288+%2B+2%2Asqrt%2825%2B3e%5E10%29%29%2F6 ALIGN=MIDDLE  ALT=\"%288+%2B+2%2Asqrt%2825%2B3e%5E10%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284+%2B+sqrt%2825%2B3e%5E10%29%29%2F3 ALIGN=MIDDLE  ALT=\"%284+%2B+sqrt%2825%2B3e%5E10%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--- in simplest form </SPAN>\n" );
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