document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=717042>Question 717042</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve for x: Log4/x(x is the base)= 5/4 + Logx/4(4 is the base). Text book answer: x = 0,334,text book author confirms value of x. Please assist. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #440066 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3759)</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=440066\">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=log%28x%2C+%284%29%29+=+5%2F4+%2B+log%284%2C+%28x%29%29 ALIGN=MIDDLE  ALT=\"log%28x%2C+%284%29%29+=+5%2F4+%2B+log%284%2C+%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In order to solve this we need to have the bases of the logarithms match. For this we will use the change of base formula, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We can either change the first log from base x to base 4, the second log from base x to base 4 or change both logs to base whatever-base-we-want. I'm going to change the first log:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%284%2C+%284%29%29%2Flog%284%2C+%28x%29%29+=+5%2F4+%2B+log%284%2C+%28x%29%29 ALIGN=MIDDLE  ALT=\"log%284%2C+%284%29%29%2Flog%284%2C+%28x%29%29+=+5%2F4+%2B+log%284%2C+%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The numerator will simplify. When the base and the argument match like this, the log is 1:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Flog%284%2C+%28x%29%29+=+5%2F4+%2B+log%284%2C+%28x%29%29 ALIGN=MIDDLE  ALT=\"1%2Flog%284%2C+%28x%29%29+=+5%2F4+%2B+log%284%2C+%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now that the bases of the logs match we can start solving. Let's eliminate the fractions first. Multiplying by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4log%284%2C+%28x%29%29 ALIGN=MIDDLE  ALT=\"4log%284%2C+%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4log%284%2C+%28x%29%29%281%2Flog%284%2C+%28x%29%29%29+=+4log%284%2C+%28x%29%29%285%2F4+%2B+log%284%2C+%28x%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the Distributive Property on the right side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4log%284%2C+%28x%29%29%281%2Flog%284%2C+%28x%29%29%29+=+4log%284%2C+%28x%29%29%285%2F4%29+%2B+4log%284%2C+%28x%29%29%28log%284%2C+%28x%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+=5log%284%2C+%28x%29%29+%2B+4%28log%284%2C+%28x%29%29%29%5E2 ALIGN=MIDDLE  ALT=\"4+=5log%284%2C+%28x%29%29+%2B+4%28log%284%2C+%28x%29%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The equation is in what is called \"quadratic form\". If you have trouble recognizing this, then a temporary variable can help:<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+log%284%2C+%28x%29%29 ALIGN=MIDDLE  ALT=\"q+=+log%284%2C+%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Substituting this into our equation we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+=+5q+%2B+4q%5E2 ALIGN=MIDDLE  ALT=\"4+=+5q+%2B+4q%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is obviously a quadratic equation. We can solve this. Subtracting 4 (and reordering the terms):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+4q%5E2+%2B+5q++-4 ALIGN=MIDDLE  ALT=\"0+=+4q%5E2+%2B+5q++-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This will not factor so we must use the Quadratic formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+%28-%285%29+%2B-+sqrt%28%285%29%5E2-4%284%29%28-4%29%29%29%2F2%284%29 ALIGN=MIDDLE  ALT=\"q+=+%28-%285%29+%2B-+sqrt%28%285%29%5E2-4%284%29%28-4%29%29%29%2F2%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+%28-%285%29+%2B-+sqrt%2825-4%284%29%28-4%29%29%29%2F2%284%29 ALIGN=MIDDLE  ALT=\"q+=+%28-%285%29+%2B-+sqrt%2825-4%284%29%28-4%29%29%29%2F2%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+%28-%285%29+%2B-+sqrt%2825%2B64%29%29%2F2%284%29 ALIGN=MIDDLE  ALT=\"q+=+%28-%285%29+%2B-+sqrt%2825%2B64%29%29%2F2%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+%28-%285%29+%2B-+sqrt%2889%29%29%2F2%284%29 ALIGN=MIDDLE  ALT=\"q+=+%28-%285%29+%2B-+sqrt%2889%29%29%2F2%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+%28-5+%2B-+sqrt%2889%29%29%2F8 ALIGN=MIDDLE  ALT=\"q+=+%28-5+%2B-+sqrt%2889%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is short for:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+%28-5+%2B+sqrt%2889%29%29%2F8 ALIGN=MIDDLE  ALT=\"q+=+%28-5+%2B+sqrt%2889%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+%28-5+-+sqrt%2889%29%29%2F8 ALIGN=MIDDLE  ALT=\"q+=+%28-5+-+sqrt%2889%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Of course we are not interested in solutions for q. We want solutions for x. So now we substitute back in for the q:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%284%2C+%28x%29%29+=+%28-5+%2B+sqrt%2889%29%29%2F8 ALIGN=MIDDLE  ALT=\"log%284%2C+%28x%29%29+=+%28-5+%2B+sqrt%2889%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%284%2C+%28x%29%29+=+%28-5+-+sqrt%2889%29%29%2F8 ALIGN=MIDDLE  ALT=\"log%284%2C+%28x%29%29+=+%28-5+-+sqrt%2889%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we solve for x. The next step is to rewrite these equations in exponential form. Before that, however, let's get decimal approximations for the right sides. Rounded to 3 places we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%284%2C+%28x%29%29+=+0.554 ALIGN=MIDDLE  ALT=\"log%284%2C+%28x%29%29+=+0.554\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%284%2C+%28x%29%29+=+-1.804 ALIGN=MIDDLE  ALT=\"log%284%2C+%28x%29%29+=+-1.804\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now to exponential form... In general <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%29%29+=+n ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%29%29+=+n\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+a%5En ALIGN=MIDDLE  ALT=\"p+=+a%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Using this pattern on our equations we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4%5E0.554 ALIGN=MIDDLE  ALT=\"x+=+4%5E0.554\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4%5E%28-1.804%29 ALIGN=MIDDLE  ALT=\"x+=+4%5E%28-1.804%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using our calculators to find these powers of 4:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2.155 ALIGN=MIDDLE  ALT=\"x+=+2.155\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0.082 ALIGN=MIDDLE  ALT=\"x+=+0.082\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Next we check out solutions. This is <i>not optional</i> when solving logarithmic equations like this. You must at least ensure that all bases and all arguments are valid. (Valid bases and arguments are positive and bases cannot be a 1.) We can quickly see that both solutions will make the the bases and arguments of the logs valid. If we had gotten an invalid base or argument, then we would reject that solution.<br><BR>\n" );
document.write( "If you want to complete the check to see if the solutions actually work (this part of the check is optional), then we would need to have logs with bases our calculators \"know\", base 10 (\"log\") or base e (\"ln\"). Using the change of base formula on the original equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%2C+%284%29%29+=+5%2F4+%2B+log%284%2C+%28x%29%29 ALIGN=MIDDLE  ALT=\"log%28x%2C+%284%29%29+=+5%2F4+%2B+log%284%2C+%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to convert the logs to base e logs we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%284%29%2Fln%28x%29+=+5%2F4+%2B+ln%28x%29%2Fln%284%29 ALIGN=MIDDLE  ALT=\"ln%284%29%2Fln%28x%29+=+5%2F4+%2B+ln%28x%29%2Fln%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then for each solution we replace the x's with that solution and use our calculators to see if it checks. (Note: Because our calculators give us decimal approximations, we may end up with a left side that is very, very close but not exactly equal to the right side. If so then we can assume that the solution checks. Both of these answers check.<br><BR>\n" );
document.write( "We have two correct answers but you say that there is just one answer (which is different from both of the ones we found above). I can only suspect that the reason for this is that there is something wrong with the equation you posted. I hope that you can use the above to help you figure out the solution(s) to the actual equation.<br><BR>\n" );
document.write( "P.S. One you have had some practice with quadratic form equations, you will no longer need a temporary variable. You will be able to see how to go directly from:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+=5log%284%2C+%28x%29%29+%2B+4%28log%284%2C+%28x%29%29%29%5E2 ALIGN=MIDDLE  ALT=\"4+=5log%284%2C+%28x%29%29+%2B+4%28log%284%2C+%28x%29%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=4%28log%284%2C+%28x%29%29%29%5E2+%2B+5log%284%2C+%28x%29%29+-+4+ ALIGN=MIDDLE  ALT=\"0+=4%28log%284%2C+%28x%29%29%29%5E2+%2B+5log%284%2C+%28x%29%29+-+4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%284%2C+%28x%29%29+=+%28-%285%29+%2B-+sqrt%28%285%29%5E2-4%284%29%28-4%29%29%29%2F2%284%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "etc. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );