document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=335748>Question 335748</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Simplify using at most a single logarithm:<BR>\n" );
document.write( "logA^(3)-logB^(2/3)+logA^(1/3)+logB^(5/3) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #240878 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=240878\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I believe this means:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28a%5E3%29%29+-+log%28%28b%5E%282%2F3%29%29%29+%2B+log%28%28a%5E%281%2F3%29%29%29+%2B+log%28%28b%5E%285%2F3%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If so, then this is the same as:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28%28a%5E3+%2A+a%5E%281%2F3%29+%2A+b%5E%285%2F3%29%29+%2F+b%282%2F3%29%29%29 ALIGN=MIDDLE  ALT=\"log%28%28%28a%5E3+%2A+a%5E%281%2F3%29+%2A+b%5E%285%2F3%29%29+%2F+b%282%2F3%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You can simplify this expression to be equal to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28a%5E%2810%2F3%29+%2A+b%29%29 ALIGN=MIDDLE  ALT=\"log%28%28a%5E%2810%2F3%29+%2A+b%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You can confirm this to be true by plugging any values for a and b and seeing if both equations provide you with the same answer.\r<BR>\n" );
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document.write( "For example, letting a = 1 and b = 2 yields the following:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28a%5E%2810%2F3%29+%2A+b%29%29 ALIGN=MIDDLE  ALT=\"log%28%28a%5E%2810%2F3%29+%2A+b%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%281%5E%2810%2F3%29+%2A+2%29%29 ALIGN=MIDDLE  ALT=\"log%28%281%5E%2810%2F3%29+%2A+2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%282%29%29 ALIGN=MIDDLE  ALT=\"log%28%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which equals .301029996\r<BR>\n" );
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document.write( "and the following:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28a%5E3%29%29+-+log%28%28b%5E%282%2F3%29%29%29+%2B+log%28%28a%5E%281%2F3%29%29%29+%2B+log%28%28b%5E%285%2F3%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > which equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%281%5E3%29%29+-+log%28%282%5E%282%2F3%29%29%29+%2B+log%28%281%5E%281%2F3%29%29%29+%2B+log%28%282%5E%285%2F3%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > which equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%281%29%29+-+log%28%281.587401052%29%29+%2B+log%28%281%29%29+%2B+log%28%283.174802104%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > which equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+-+.200686664+%2B+0+%2B+.501716659 ALIGN=MIDDLE  ALT=\"0+-+.200686664+%2B+0+%2B+.501716659\"   DISABLED_style= \"max-width:90%; height:auto;\" > which equals .301029996\r<BR>\n" );
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document.write( "Since you get the same answer with both equations, then the simplification is confirmed as good.\r<BR>\n" );
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document.write( "You used the following concepts to solve this.\r<BR>\n" );
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document.write( "log(x) + log(y) = log(x*y)\r<BR>\n" );
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document.write( "log(x) - log(y) = log(x/y)\r<BR>\n" );
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document.write( "In particular:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28a%5E3%29%29+%2B+log%28%28a%5E%281%2F3%29%29%29+%2B+log%28%28b%5E%285%2F3%29%29%29 ALIGN=MIDDLE  ALT=\"log%28%28a%5E3%29%29+%2B+log%28%28a%5E%281%2F3%29%29%29+%2B+log%28%28b%5E%285%2F3%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28a%5E3%2Aa%281%2F3%29%2Ab%285%2F3%29%29%29 ALIGN=MIDDLE  ALT=\"log%28%28a%5E3%2Aa%281%2F3%29%2Ab%285%2F3%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Also in particular:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28a%5E3%2Aa%281%2F3%29%2Ab%285%2F3%29%29%29+-+log%28%28b%5E%282%2F3%29%29%29 ALIGN=MIDDLE  ALT=\"log%28%28a%5E3%2Aa%281%2F3%29%2Ab%285%2F3%29%29%29+-+log%28%28b%5E%282%2F3%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28%28a%5E3%2Aa%5E%281%2F3%29%2Ab%285%2F3%29%29%2Fb%282%2F3%29%29%29%29 ALIGN=MIDDLE  ALT=\"log%28%28%28a%5E3%2Aa%5E%281%2F3%29%2Ab%285%2F3%29%29%2Fb%282%2F3%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Some more concepts:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Em+%2F+x%5En+=+x%5E%28m-n%29 ALIGN=MIDDLE  ALT=\"x%5Em+%2F+x%5En+=+x%5E%28m-n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In particular:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E%283%29+%2A+a%5E%281%2F3%29+=+a%5E%283+%2B+1%2F3%29+=+a%5E%2810%2F3%29 ALIGN=MIDDLE  ALT=\"a%5E%283%29+%2A+a%5E%281%2F3%29+=+a%5E%283+%2B+1%2F3%29+=+a%5E%2810%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Also in particular:\r<BR>\n" );
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document.write( " </SPAN>\n" );
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