document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=415395>Question 415395</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is this. Find x,y and z<BR>\n" );
document.write( "log3(log4(log5x))=log4(log5(log3y))=log5(log3(log4z))=0\r<BR>\n" );
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document.write( "The numbers, log3<-...log4<-...log5<-, are bases. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #291334 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=291334\">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%283%2C+%28log%284%2C+%28log%285%2C+%28x%29%29%29%29%29%29+=+log%284%2C+%28log%285%2C+%28log%283%2C+%28y%29%29%29%29%29%29+=+log%285%2C+%28log%283%2C+%28log%284%2C+%28z%29%29%29%29%29%29+=+0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "First of all we need to understand what this \"continued\" equation means and how to work with them. The notation above is just a shorthand way of saying that any combination of 2 of these 4 expressions are equal to each other. There are 6 combinations of 2 of these four expressions so the notation above saves us from having to write 8 different \"regular\" equations (one expression equals another expression).<br><BR>\n" );
document.write( "To work with this notation one usualy will pick two of the expressions, write a \"regular\" equation for those two expressions and use regular Algebra on that equation. To find x, y and z we will \"extract\" and use the following equations, respectively:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%283%2C+%28log%284%2C+%28log%285%2C+%28x%29%29%29%29%29%29+=+0 ALIGN=MIDDLE  ALT=\"log%283%2C+%28log%284%2C+%28log%285%2C+%28x%29%29%29%29%29%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%284%2C+%28log%285%2C+%28log%283%2C+%28y%29%29%29%29%29%29+=+0 ALIGN=MIDDLE  ALT=\"log%284%2C+%28log%285%2C+%28log%283%2C+%28y%29%29%29%29%29%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%285%2C+%28log%283%2C+%28log%284%2C+%28z%29%29%29%29%29%29+=+0 ALIGN=MIDDLE  ALT=\"log%285%2C+%28log%283%2C+%28log%284%2C+%28z%29%29%29%29%29%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We use these 3 (of the six) because these are the ones that have just one variable in them.<br><BR>\n" );
document.write( "To solve each of these equations we will \"peel away\" the logarithms, one-by-one, by rewriting the equaiton in exponential form repeatedly. I will do one of the three and from that you should be able to figure out the other two.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%283%2C+%28log%284%2C+%28log%285%2C+%28x%29%29%29%29%29%29+=+0 ALIGN=MIDDLE  ALT=\"log%283%2C+%28log%284%2C+%28log%285%2C+%28x%29%29%29%29%29%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In general, the logarithmic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%29%29+=+q ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%29%29+=+q\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+a%5Eq ALIGN=MIDDLE  ALT=\"p+=+a%5Eq\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Using this pattern on the outermost log (the base 3 log) we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%284%2C+%28log%285%2C+%28x%29%29%29%29+=+3%5E0 ALIGN=MIDDLE  ALT=\"log%284%2C+%28log%285%2C+%28x%29%29%29%29+=+3%5E0\"   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%284%2C+%28log%285%2C+%28x%29%29%29%29+=+1 ALIGN=MIDDLE  ALT=\"log%284%2C+%28log%285%2C+%28x%29%29%29%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Repeat:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%285%2C+%28x%29%29+=+4%5E1 ALIGN=MIDDLE  ALT=\"log%285%2C+%28x%29%29+=+4%5E1\"   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%285%2C+%28x%29%29+=+4 ALIGN=MIDDLE  ALT=\"log%285%2C+%28x%29%29+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Repeat:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5%5E4 ALIGN=MIDDLE  ALT=\"x+=+5%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "x = 625<br><BR>\n" );
document.write( "Repeat this process with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%284%2C+%28log%285%2C+%28log%283%2C+%28y%29%29%29%29%29%29+=+0 ALIGN=MIDDLE  ALT=\"log%284%2C+%28log%285%2C+%28log%283%2C+%28y%29%29%29%29%29%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%285%2C+%28log%283%2C+%28log%284%2C+%28z%29%29%29%29%29%29+=+0 ALIGN=MIDDLE  ALT=\"log%285%2C+%28log%283%2C+%28log%284%2C+%28z%29%29%29%29%29%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );