document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=149609>Question 149609</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How would you solve: log(base 2x)+log(base 4x)+log(base 8x) = 1? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #109750 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=109750\">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%282%2C%28x%29%29%2Blog%284%2C%28x%29%29%2Blog%288%2C%28x%29%29=1 ALIGN=MIDDLE  ALT=\"log%282%2C%28x%29%29%2Blog%284%2C%28x%29%29%2Blog%288%2C%28x%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%29%29%2Flog%2810%2C%282%29%29%2Blog%2810%2C%28x%29%29%2Flog%2810%2C%284%29%29%2Blog%2810%2C%28x%29%29%2Flog%2810%2C%288%29%29=1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Use the Change of Base formula to rewrite each log. Remember, the Change of Base formula is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C%28x%29%29=log%2810%2C%28x%29%29%2Flog%2810%2C%28b%29%29 ALIGN=MIDDLE  ALT=\"log%28b%2C%28x%29%29=log%2810%2C%28x%29%29%2Flog%2810%2C%28b%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%29%29%2Flog%2810%2C%282%29%29%2Blog%2810%2C%28x%29%29%2Flog%2810%2C%282%5E2%29%29%2Blog%2810%2C%28x%29%29%2Flog%2810%2C%282%5E3%29%29=1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E2 ALIGN=MIDDLE  ALT=\"2%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E3 ALIGN=MIDDLE  ALT=\"2%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%29%29%2Flog%2810%2C%282%29%29%2Blog%2810%2C%28x%29%29%2F%282%2Alog%2810%2C%282%29%29%29%2Blog%2810%2C%28x%29%29%2F%283%2Alog%2810%2C%282%29%29%29=1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite each log using the identity  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29 ALIGN=MIDDLE  ALT=\"log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now to make things simple, let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=log%2810%2C%282%29%29 ALIGN=MIDDLE  ALT=\"z=log%2810%2C%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this means that the equation is now\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%29%29%2Fz%2Blog%2810%2C%28x%29%29%2F%282z%29%2Blog%2810%2C%28x%29%29%2F%283z%29=1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Notice now that the LCD is 6z\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6z%28log%2810%2C%28x%29%29%2Fcross%28z%29%2Blog%2810%2C%28x%29%29%2Fcross%282z%29%2Blog%2810%2C%28x%29%29%2Fcross%283z%29%29=6z%281%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by the LCD <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6z ALIGN=MIDDLE  ALT=\"6z\"   DISABLED_style= \"max-width:90%; height:auto;\" > to clear the fractions.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2Alog%2810%2C%28x%29%29%2B3%2Alog%2810%2C%28x%29%29%2B2%2Alog%2810%2C%28x%29%29=6z ALIGN=MIDDLE  ALT=\"6%2Alog%2810%2C%28x%29%29%2B3%2Alog%2810%2C%28x%29%29%2B2%2Alog%2810%2C%28x%29%29=6z\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute and multiply.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%29%29%286%2B3%2B2%29=6z ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%286%2B3%2B2%29=6z\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%29%29%2811%29=6z ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%2811%29=6z\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2Alog%2810%2C%28x%29%29=6z ALIGN=MIDDLE  ALT=\"11%2Alog%2810%2C%28x%29%29=6z\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%29%29=%286z%29%2F%2811%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29=%286z%29%2F%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 11.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%29%29=%286%2Alog%2810%2C%282%29%29%29%2F%2811%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29=%286%2Alog%2810%2C%282%29%29%29%2F%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now replace \"z\" with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%282%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%29%29=%286%2F11%29%2Alog%2810%2C%282%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29=%286%2F11%29%2Alog%2810%2C%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%29%29%2Flog%2810%2C%282%29%29=6%2F11 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%2Flog%2810%2C%282%29%29=6%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%282%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C%28x%29%29=6%2F11 ALIGN=MIDDLE  ALT=\"log%282%2C%28x%29%29=6%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the change of base formula to rewrite the left side.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2^{\frac{6}{11}}=x\"> Rewrite the equation using the property: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C%28x%29%29=y ALIGN=MIDDLE  ALT=\"log%28b%2C%28x%29%29=y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ====> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Ey=x ALIGN=MIDDLE  ALT=\"b%5Ey=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the answer is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x=2^{\frac{6}{11}}\"> which approximates to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1.45948 ALIGN=MIDDLE  ALT=\"x=1.45948\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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