document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1043663>Question 1043663</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if log(a+b+c)=loga+logb+logc then prove that log(2a/1-a^2 + 2b/1-b^2 + 2c/1-c^2)= log 2a/1-a^2 + log 2b/1-b^2 + log 2c/1-c^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #660462 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=660462\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We wish to show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28a%2Bb%2Bc%29%29=loga%2Blogb%2Blogc ALIGN=MIDDLE  ALT=\"log%28%28a%2Bb%2Bc%29%29=loga%2Blogb%2Blogc\"   DISABLED_style= \"max-width:90%; height:auto;\" > implies that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%282a%2F%281-a%5E2%29+%2B+2b%2F%281-b%5E2%29+%2B+2c%2F%281-c%5E2%29%29%29=+log+%28%282a%2F%281-a%5E2%29%29%29+%2B+%0D%0Alog+%28%282b%2F%281-b%5E2%29%29%29+%2B+log+%28%282c%2F%281-c%5E2%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "For the result to have meaning, it must be that 0 < a,b,c < 1, which can be determined by solving the inequalities<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%2F%281-a%5E2%29%3E0 ALIGN=MIDDLE  ALT=\"2a%2F%281-a%5E2%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b%2F%281-b%5E2%29%3E0 ALIGN=MIDDLE  ALT=\"2b%2F%281-b%5E2%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2c%2F%281-c%5E2%29%3E0 ALIGN=MIDDLE  ALT=\"2c%2F%281-c%5E2%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "With this in mind, it is enough to show that \r<BR>\n" );
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document.write( "a+b+c = abc  ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2F%281-a%5E2%29+%2B+b%2F%281-b%5E2%29+%2B+c%2F%281-c%5E2%29+=+%0D%0A%284abc%29%2F%28%281-a%5E2%29%281-b%5E2%29%281-c%5E2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.  <---- Why?\r<BR>\n" );
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document.write( "Now on to the proof:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28ab%2Bac%2Bbc%29abc+%2B+%28ab%2Bbc%2Bac%29abc+=+0 ALIGN=MIDDLE  ALT=\"-%28ab%2Bac%2Bbc%29abc+%2B+%28ab%2Bbc%2Bac%29abc+=+0\"   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=-%28ab%2Bac%2Bbc%29%28a%2Bb%2Bc%29+%2B+%28ab%2Bbc%2Bac%29abc+=+0 ALIGN=MIDDLE  ALT=\"-%28ab%2Bac%2Bbc%29%28a%2Bb%2Bc%29+%2B+%28ab%2Bbc%2Bac%29abc+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, since a+b+c = abc,\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-ab%28a%2Bb%2Bc%29+-+ac%28a%2Bb%2Bc%29-+bc%28a%2Bb%2Bc%29+%2B+%28ab%2Bbc%2Bac%29abc+=+0 ALIGN=MIDDLE  ALT=\"-ab%28a%2Bb%2Bc%29+-+ac%28a%2Bb%2Bc%29-+bc%28a%2Bb%2Bc%29+%2B+%28ab%2Bbc%2Bac%29abc+=+0\"   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=-ab%28a%2Bb%2Bc%29+%2B+abc+-+ac%28a%2Bb%2Bc%29+%2B+abc+-+bc%28a%2Bb%2Bc%29+%2B+abc+%2B++%28ab%2Bbc%2Bac%29abc=+3abc ALIGN=MIDDLE   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=-ab%28a%2Bb%29+-ac%28a%2Bc%29-bc%28b%2Bc%29+%2B+%28ab%2Bbc%2Bac%29abc+=+3abc ALIGN=MIDDLE  ALT=\"-ab%28a%2Bb%29+-ac%28a%2Bc%29-bc%28b%2Bc%29+%2B+%28ab%2Bbc%2Bac%29abc+=+3abc\"   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=-a%5E2b+-+ab%5E2+-+a%5E2c+-+ac%5E2+-+b%5E2c-bc%5E2+%2B+%28ab%2Bbc%2Bac%29abc+=+3abc ALIGN=MIDDLE  ALT=\"-a%5E2b+-+ab%5E2+-+a%5E2c+-+ac%5E2+-+b%5E2c-bc%5E2+%2B+%28ab%2Bbc%2Bac%29abc+=+3abc\"   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=-+ab%5E2-+ac%5E2+-+a%5E2b+-+bc%5E2++-+a%5E2c+-+b%5E2c++%2B+%28ab%2Bbc%2Bac%29abc+=+3abc ALIGN=MIDDLE  ALT=\"-+ab%5E2-+ac%5E2+-+a%5E2b+-+bc%5E2++-+a%5E2c+-+b%5E2c++%2B+%28ab%2Bbc%2Bac%29abc+=+3abc\"   DISABLED_style= \"max-width:90%; height:auto;\" > after a little rearrangement of terms.\r<BR>\n" );
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document.write( "Since a+b+c = abc, we get \r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-+ab%5E2-+ac%5E2%29%2B+%28b+-+a%5E2b+-+bc%5E2%29+%2B%28c+-+a%5E2c+-+b%5E2c%29++%2B+%28ab%2Bbc%2Bac%29abc+=+4abc ALIGN=MIDDLE   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=%28a-+ab%5E2-+ac%5E2+%2B+ab%5E2c%5E2%29%2B+%28b+-+a%5E2b+-+bc%5E2+%2B+ba%5E2c%5E2%29+%2B%28c+-+a%5E2c+-+b%5E2c+%2B+ca%5E2b%5E2%29+=+4abc ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281-+b%5E2-+c%5E2+%2B+b%5E2c%5E2%29%2B+b%281+-+a%5E2+-+c%5E2+%2B+a%5E2c%5E2%29+%2B+c%281+-+a%5E2+-+b%5E2+%2B+a%5E2b%5E2%29+=+4abc ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281-b%5E2%29%281-c%5E2%29+%2B+b%281-a%5E2%29%281-c%5E2%29+%2B+c%281-a%5E2%29%281-b%5E2%29+=+4abc ALIGN=MIDDLE   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=a%2F%281-a%5E2%29+%2B+b%2F%281-b%5E2%29+%2B+c%2F%281-c%5E2%29+=+%284abc%29%2F%28%281-a%5E2%29%281-b%5E2%29%281-c%5E2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, after dividing both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%281-a%5E2%29%281-b%5E2%29%281-c%5E2%29 ALIGN=MIDDLE  ALT=\"+%281-a%5E2%29%281-b%5E2%29%281-c%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And that's it...\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );