document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=283809>Question 283809</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2log(7,3) + 3log(7,2) = log(7, x) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #205953 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=205953\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 2*log(7,3) + 3*log(7,2) = log(7,x)\r<BR>\n" );
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document.write( "We can rewrite this as:\r<BR>\n" );
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document.write( "log(7,x) = 2*log(7,3) + 3*log(7,2)\r<BR>\n" );
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document.write( "The rules of logarithms that apply here will be:\r<BR>\n" );
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document.write( "log(a^n) = n*log(a)<BR>\n" );
document.write( "log(a*b) = log(a) + log(b)\r<BR>\n" );
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document.write( "First thing we notice is that 2*log(7,3) would be the same as log(7,3^2) which would be the same as log(7,9).\r<BR>\n" );
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document.write( "Second thing we notice is that 3*log(7,2) would be the same as log(7,2^3) which would be the same as log(7,8).\r<BR>\n" );
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document.write( "Our equation becomes:\r<BR>\n" );
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document.write( "log(7,x) = log(7,9) + log(7,8)\r<BR>\n" );
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document.write( "Third thing we notice is that log(7,9) + log(7,8) would be the same as log(7,9*8) which would be the same as log(7,72)\r<BR>\n" );
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document.write( "Our equation becomes:\r<BR>\n" );
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document.write( "log(7,x) = log(7,72)\r<BR>\n" );
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document.write( "This can only be true if x = 72.\r<BR>\n" );
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document.write( "Our original equation is:\r<BR>\n" );
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document.write( "log(7,x) = 2*log(7,3) + 3*log(7,2)\r<BR>\n" );
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document.write( "We replace x with 72 and we get:\r<BR>\n" );
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document.write( "log(7,72) = 2*log(7,3) + 3*log(7,2)\r<BR>\n" );
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document.write( "We can convert this equation to the base of 10 so we can solve using our calculator.\r<BR>\n" );
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document.write( "The conversion formula in general is:\r<BR>\n" );
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document.write( "log(b,x) = log(c,x)/log(c,b)\r<BR>\n" );
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document.write( "This reads as log of x to the base b = log of x to the base c divided by the log of b to the base c.\r<BR>\n" );
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document.write( "The conversion formula in our case is:\r<BR>\n" );
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document.write( "log(7,x) = log(10,x) / log(10,7)\r<BR>\n" );
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document.write( "Our equation becomes:\r<BR>\n" );
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document.write( "log(10,72/log(10,7) = 2*log(10,3)/log(10,7) + 3*log(10,2)/log(10,7)\r<BR>\n" );
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document.write( "If we multiply both sides of this equation by log(10,7), then we get:\r<BR>\n" );
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document.write( "log(10,72) = 2*log(10,3) + 3*log(10,2) which becomes:\r<BR>\n" );
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document.write( "1.857332496 = 2*.477121255 + 3*.301029996 which becomes:\r<BR>\n" );
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document.write( "1.857332496 = 1.857332496 which is true confirming the equations are equivalent.\r<BR>\n" );
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document.write( "Your answer is that x = 72.\r<BR>\n" );
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