document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=492071>Question 492071</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the exact value for the expression, log5 270 &#8722; log5 75 &#8722; log5 90 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #334933 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=334933\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your expression is:\r<BR>\n" );
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document.write( "log(5,270) - log(5,75) - log(5,90)\r<BR>\n" );
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document.write( "in general:\r<BR>\n" );
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document.write( "log(a/b) = log(a) - log(b) and:<BR>\n" );
document.write( "log(a) - log(b) = log(a/b)\r<BR>\n" );
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document.write( "your expression gets transformed using this property as follows:\r<BR>\n" );
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document.write( "log(5,270) - log(5,75) - log(5,90)\r<BR>\n" );
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document.write( "becomes:\r<BR>\n" );
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document.write( "log(5,(270/75/90) which becomes:\r<BR>\n" );
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document.write( "log(5,.04)\r<BR>\n" );
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document.write( "set y equal to it to get the equation:\r<BR>\n" );
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document.write( "y = log(5,.04)\r<BR>\n" );
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document.write( "you can convert this to base 10 by using the following formula:\r<BR>\n" );
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document.write( "log(5,.04) = log(10,.04) / log(10,5)\r<BR>\n" );
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document.write( "now you can use the LOG function of your calculator to solve.\r<BR>\n" );
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document.write( "using your calculation, you get y = log(5,.04) becomes y = log(10,.04) / log(10,5) = -2\r<BR>\n" );
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document.write( "what this says is that y = log(5,.04) = -2.\r<BR>\n" );
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document.write( "the basic properties of logarithms says that:\r<BR>\n" );
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document.write( "log(b,x) = y if and only if b^y = x\r<BR>\n" );
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document.write( "with your equation, this becomes:\r<BR>\n" );
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document.write( "log(5,.04) = -2 if and only if 5^(-2) = .04\r<BR>\n" );
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document.write( "5^(-2) is the same as 1/5^2 which is equal to 1/25 which is equal to .04.\r<BR>\n" );
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document.write( "this confirms the fact that y = -2 is your answer.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "log(5,270) - log(5,75) - log(5,90) = -2\r<BR>\n" );
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document.write( "if you don't do anything except convert log(5,x) to log(10,x) /log(10,5), then you should get the same answer.\r<BR>\n" );
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document.write( "your expression becomes:\r<BR>\n" );
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document.write( "log(10,270)/log(10,5) - log(10,75)/log(10,5) - log(10,90)/log(10,5) = -2 which becomes:\r<BR>\n" );
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document.write( "3.478495142 - 2.682606194 - 2.795888947 = -2.\r<BR>\n" );
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document.write( "you get -2 again, confirming that -2 is the correct answer.\r<BR>\n" );
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document.write( "you can convert a log from any base to any other base by using the conversion formula.\r<BR>\n" );
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document.write( "the general form of the conversion formula 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 converts a log from the base of b to the base of c.\r<BR>\n" );
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document.write( "you make the log of x to the base of b equal to:<BR>\n" );
document.write( "the log of x to the base of c divided by:<BR>\n" );
document.write( "the log of b to the base of c.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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