document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=332329>Question 332329</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help me solve:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2+%2A+log%284%2C%28x%29%29+%2B+3+%2A+log%288%2C%28x%29%29+=+10 ALIGN=MIDDLE  ALT=\"+2+%2A+log%284%2C%28x%29%29+%2B+3+%2A+log%288%2C%28x%29%29+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #238292 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=238292\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Logarithmic equations are usually easier to solve when the bases of all the logarithms are the same. This allows one to use the properties of logarithms.  So the simplest solution to this equation probably starts with changing the bases of the logarithms, using the change-of-base formula. 4 and 8 are not (obvious) powers of each other but they are both powers of 2 so we change both logarithms in base 2 logarithms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2+%2A+log%282%2C%28x%29%29%2Flog%282%2C+%284%29%29+%2B+3+%2A+log%282%2C%28x%29%29%2Flog%282%2C+%288%29%29+=+10 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E2+=+4 ALIGN=MIDDLE  ALT=\"2%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C+%284%29%29+=+2 ALIGN=MIDDLE  ALT=\"log%282%2C+%284%29%29+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Similarly, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C+%288%29%29+=+3 ALIGN=MIDDLE  ALT=\"log%282%2C+%288%29%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Substituting we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2+%2A+log%282%2C%28x%29%29%2F2+%2B+3+%2A+log%282%2C%28x%29%29%2F3+=+10 ALIGN=MIDDLE  ALT=\"+2+%2A+log%282%2C%28x%29%29%2F2+%2B+3+%2A+log%282%2C%28x%29%29%2F3+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+log%282%2C%28x%29%29+%2B+log%282%2C%28x%29%29+=+10 ALIGN=MIDDLE  ALT=\"+log%282%2C%28x%29%29+%2B+log%282%2C%28x%29%29+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now that the bases are the same we can use a property of logarithms to combine the remaining logs:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C+%28x%2Ax%29%29+=+10 ALIGN=MIDDLE  ALT=\"log%282%2C+%28x%2Ax%29%29+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C+%28x%5E2%29%29+=+10 ALIGN=MIDDLE  ALT=\"log%282%2C+%28x%5E2%29%29+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we can write this in exponential form:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E10+=+x%5E2 ALIGN=MIDDLE  ALT=\"2%5E10+=+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solving this equation we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E5+=+x ALIGN=MIDDLE  ALT=\"2%5E5+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%5E5+=+x ALIGN=MIDDLE  ALT=\"-2%5E5+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32+=+x ALIGN=MIDDLE  ALT=\"32+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-32+=+x ALIGN=MIDDLE  ALT=\"-32+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "When solving logarithmic equations you must check your answers. You must reject any \"solutions\" which would cause the argument of a logarithm to become negative. We can easily see that the arguments of both logarithms of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2+%2A+log%284%2C%28x%29%29+%2B+3+%2A+log%288%2C%28x%29%29+=+10 ALIGN=MIDDLE  ALT=\"+2+%2A+log%284%2C%28x%29%29+%2B+3+%2A+log%288%2C%28x%29%29+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "will become negative if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-32+=+x ALIGN=MIDDLE  ALT=\"-32+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So we must reject this \"solution\". (If only one argument becomes negative we would still reject the solution.)<br><BR>\n" );
document.write( "So the only solution to your equation is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32+=+x ALIGN=MIDDLE  ALT=\"32+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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