document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=222790>Question 222790</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let (((log(a,X))))=C and (((log(b,X))))=D. Find the general statement that expresses (((log(ab,X))))in term of C and D.\r<BR>\n" );
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document.write( "I already know the answer is (CD)/(C+D)\r<BR>\n" );
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document.write( "I cannot figure out how to write a proof for this question. \r<BR>\n" );
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document.write( "I know all of the logarithmic principles as well but I  can only come up with either (C)(D) as my answer or (C+D). I can't get a fraction.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #166740 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=166740\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I found that it helps to work backwards for a bit to find out how to go forward. In this case, I broke down <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28ab%2C%28x%29%29 ALIGN=MIDDLE  ALT=\"log%28ab%2C%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:\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%28ab%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%2Flog%2810%2C%28ab%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (used the change of base formula here)\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%2F%28log%2810%2C%28a%29%29%2Blog%2810%2C%28b%29%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%2F%28log%2810%2C%28a%29%29%2Blog%2810%2C%28b%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (used the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C%28A%2AB%29%29=log%28b%2C%28A%29%29%2Blog%28b%2C%28B%29%29 ALIGN=MIDDLE  ALT=\"log%28b%2C%28A%2AB%29%29=log%28b%2C%28A%29%29%2Blog%28b%2C%28B%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Then I realized that I needed to find substitutions for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28a%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28b%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28b%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (to somehow get 'x' in there). So I figured I'd solve for those respective expressions like so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C%28x%29%29=C ALIGN=MIDDLE  ALT=\"log%28a%2C%28x%29%29=C\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the first 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%28a%29%29=C ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%2Flog%2810%2C%28a%29%29=C\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the change of base formula.\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=C%2Alog%2810%2C%28a%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29=C%2Alog%2810%2C%28a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28a%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28a%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%2FC=log%2810%2C%28a%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%2FC=log%2810%2C%28a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by C.\r<BR>\n" );
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document.write( "So this means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28a%29%29=log%2810%2C%28x%29%29%2FC ALIGN=MIDDLE  ALT=\"log%2810%2C%28a%29%29=log%2810%2C%28x%29%29%2FC\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "---------------------\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C%28x%29%29=D ALIGN=MIDDLE  ALT=\"log%28b%2C%28x%29%29=D\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the second 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%28b%29%29=D ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%2Flog%2810%2C%28b%29%29=D\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the change of base formula.\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=D%2Alog%2810%2C%28b%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29=D%2Alog%2810%2C%28b%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28b%29%29 ALIGN=MIDDLE  ALT=\"log%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%2FD=log%2810%2C%28b%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%2FD=log%2810%2C%28b%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by D.\r<BR>\n" );
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document.write( "So this means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28b%29%29=log%2810%2C%28x%29%29%2FD ALIGN=MIDDLE  ALT=\"log%2810%2C%28b%29%29=log%2810%2C%28x%29%29%2FD\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "-------------------------------------------\r<BR>\n" );
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document.write( "Now let's get back to the main problem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28ab%2C%28x%29%29 ALIGN=MIDDLE  ALT=\"log%28ab%2C%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression.\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%28ab%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%2Flog%2810%2C%28ab%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the change of base formula.\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%2F%28log%2810%2C%28a%29%29%2Blog%2810%2C%28b%29%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%2F%28log%2810%2C%28a%29%29%2Blog%2810%2C%28b%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C%28A%2AB%29%29=log%28b%2C%28A%29%29%2Blog%28b%2C%28B%29%29 ALIGN=MIDDLE  ALT=\"log%28b%2C%28A%2AB%29%29=log%28b%2C%28A%29%29%2Blog%28b%2C%28B%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to break up the denominator.\r<BR>\n" );
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document.write( "Now here's where the substitutions come into play:\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%2F%28log%2810%2C%28x%29%29%2FC%2Blog%2810%2C%28x%29%29%2FD%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%2F%28log%2810%2C%28x%29%29%2FC%2Blog%2810%2C%28x%29%29%2FD%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28a%29%29=log%2810%2C%28x%29%29%2FC ALIGN=MIDDLE  ALT=\"log%2810%2C%28a%29%29=log%2810%2C%28x%29%29%2FC\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28b%29%29=log%2810%2C%28x%29%29%2FD ALIGN=MIDDLE  ALT=\"log%2810%2C%28b%29%29=log%2810%2C%28x%29%29%2FD\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From here, it's just boils down to algebraically simplifying the expression:\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%2F%28%28D%2Alog%2810%2C%28x%29%29%29%2F%28CD%29%2Blog%2810%2C%28x%29%29%2FD%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the first inner fraction by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%2FD ALIGN=MIDDLE  ALT=\"D%2FD\"   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%2F%28%28D%2Alog%2810%2C%28x%29%29%29%2F%28CD%29%2B%28C%2Alog%2810%2C%28x%29%29%29%2F%28CD%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the second inner fraction by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2FC ALIGN=MIDDLE  ALT=\"C%2FC\"   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%2F%28%28D%2Alog%2810%2C%28x%29%29%2BC%2Alog%2810%2C%28x%29%29%29%2F%28CD%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the lower fractions.\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%2A%28%28CD%29%2F%28D%2Alog%2810%2C%28x%29%29%2BC%2Alog%2810%2C%28x%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <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;\" > by the reciprocal of the lower fraction.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28CD%2Alog%2810%2C%28x%29%29%29%2F%28D%2Alog%2810%2C%28x%29%29%2BC%2Alog%2810%2C%28x%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28CD%2Alog%2810%2C%28x%29%29%29%2F%28log%2810%2C%28x%29%29%28D%2BC%29%29 ALIGN=MIDDLE  ALT=\"%28CD%2Alog%2810%2C%28x%29%29%29%2F%28log%2810%2C%28x%29%29%28D%2BC%29%29\"   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;\" > from the denominator.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28CD%2Across%28%28log%2810%2C%28x%29%29%29%29%29%2F%28cross%28%28log%2810%2C%28x%29%29%29%29%28D%2BC%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Cancel out the common terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28CD%29%2F%28D%2BC%29 ALIGN=MIDDLE  ALT=\"%28CD%29%2F%28D%2BC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28CD%29%2F%28C%2BD%29 ALIGN=MIDDLE  ALT=\"%28CD%29%2F%28C%2BD%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms (trivial, but I figured that I'd match the book).\r<BR>\n" );
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document.write( "So this means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28ab%2C%28x%29%29=%28CD%29%2F%28C%2BD%29 ALIGN=MIDDLE  ALT=\"log%28ab%2C%28x%29%29=%28CD%29%2F%28C%2BD%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C%28x%29%29=C ALIGN=MIDDLE  ALT=\"log%28a%2C%28x%29%29=C\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C%28x%29%29=D ALIGN=MIDDLE  ALT=\"log%28b%2C%28x%29%29=D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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