document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=253720>Question 253720</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a=log(base7)(11-6&#8730;2) and b=log(base7)(45+29&#8730;2), find 3a+2b in the simplest form. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #186175 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=186175\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=log%287%2C%2811-6%2Asqrt%282%29%29%29 ALIGN=MIDDLE  ALT=\"a=log%287%2C%2811-6%2Asqrt%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=log%287%2C%2845%2B29%2Asqrt%282%29%29%29 ALIGN=MIDDLE  ALT=\"b=log%287%2C%2845%2B29%2Asqrt%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, find 3a+2b in the simplest form.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+log%287%2C%2811-6%2Asqrt%282%29%29%29 ALIGN=MIDDLE  ALT=\"a+=+log%287%2C%2811-6%2Asqrt%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5Ea+=+11-6%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"7%5Ea+=+11-6%2Asqrt%282%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=b+=+log%287%2C%2845%2B29%2Asqrt%282%29%29%29 ALIGN=MIDDLE  ALT=\"b+=+log%287%2C%2845%2B29%2Asqrt%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5Eb+=+45%2B29%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"7%5Eb+=+45%2B29%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "take log of both sides of first equation to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5Ea+=+11-6%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"7%5Ea+=+11-6%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%287%5Ea%29%29+=+log%28%2811-6%2Asqrt%282%29%29%29 ALIGN=MIDDLE  ALT=\"log%28%287%5Ea%29%29+=+log%28%2811-6%2Asqrt%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Alog%28%287%29%29+=+log%28%2811-6%2Asqrt%282%29%29%29 ALIGN=MIDDLE  ALT=\"a%2Alog%28%287%29%29+=+log%28%2811-6%2Asqrt%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "divide both sides by log((7)) to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+log%28%2811-6%2Asqrt%282%29%29%29%2Flog%28%287%29%29 ALIGN=MIDDLE  ALT=\"a+=+log%28%2811-6%2Asqrt%282%29%29%29%2Flog%28%287%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > *************\r<BR>\n" );
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document.write( "take log of both sides of second equation to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5Eb+=+45%2B29%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"7%5Eb+=+45%2B29%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%287%5Eb%29%29+=+log%28%2845+%2B+29%2Asqrt%282%29%29%29 ALIGN=MIDDLE  ALT=\"log%28%287%5Eb%29%29+=+log%28%2845+%2B+29%2Asqrt%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Alog%28%287%29%29+=+log%28%2845+%2B+29%2Asqrt%282%29%29%29 ALIGN=MIDDLE  ALT=\"b%2Alog%28%287%29%29+=+log%28%2845+%2B+29%2Asqrt%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "divide both sides by log((7)) to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+log%28%2845%2B29%2Asqrt%282%29%29%29%2Flog%28%287%29%29 ALIGN=MIDDLE  ALT=\"b+=+log%28%2845%2B29%2Asqrt%282%29%29%29%2Flog%28%287%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=3a+%2B+2b+=+3%2Alog%28%2811-6%2Asqrt%282%29%29%29%2Flog%28%287%29%29+%2B+2%2Alog%28%2845%2B29%2Asqrt%282%29%29%29%2Flog%28%287%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "since the denominator is the same, we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a+%2B+2b+=+%283%2Alog%28%2811-6%2Asqrt%282%29%29%29+%2B+2%2Alog%28%2845%2B29%2Asqrt%282%29%29%29%29%2Flog%28%287%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I don't think you can simplify it any further, but you can solve it using the LOG function of your calculator.\r<BR>\n" );
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document.write( "equation becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a+%2B+2b+=+%283%2A.400489398+%2B+2%2A1.934560022%29%2F.84509804 ALIGN=MIDDLE  ALT=\"3a+%2B+2b+=+%283%2A.400489398+%2B+2%2A1.934560022%29%2F.84509804\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a+%2B+2b+=+%281.201468195+%2B+3.869120045%29%2F.84509804+=+6 ALIGN=MIDDLE  ALT=\"3a+%2B+2b+=+%281.201468195+%2B+3.869120045%29%2F.84509804+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I confirmed the answer is correct by using the logarithm base conversion formula to convert base 7 logarithm to a base 10 logarithm.\r<BR>\n" );
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document.write( "that conversion formula is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%287%2C%28x%29%29+=+log%2810%2C%28x%29%29%2Flog%2810%2C%287%29%29 ALIGN=MIDDLE  ALT=\"log%287%2C%28x%29%29+=+log%2810%2C%28x%29%29%2Flog%2810%2C%287%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "you could also have solved this problem directly by using the base conversion formula.   <BR>\n" );
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