document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=250873>Question 250873</A>: <I><SPAN STYLE=\"word-break: break-all;\"> log(x-1)+log(5x)=2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #182732 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=182732\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> log(x-1) + log(5x) = 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "three basic rules of logarithms are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(a*b) = log(a) + log(b) (rule 1)<BR>\n" );
document.write( "log(a/b) = log(a) - log(b) (rule 2)<BR>\n" );
document.write( "log(a^b) = b*log(a) (rule 3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your equation can be made into the following by invoking rule 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(x-1) + log(5x) = 2 becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log((x-1)*(5x)) = 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(5x^2 - 5x) = 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the basic rule of logarithms states:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = log(a,x) if and only if a^y = x\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in your equation, a base of 10 is implied.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your equation of log(5x^2 - 5x) = 2 is really:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(10,(5x^2-5x)) = 2 which means:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log of (5x^2-5x) to the base of 10 = 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "using the basic rule of logarithms, this equation becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(10,(5x^2-5x)) = 2 if and only if:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "10^2 = 5x^2-5x\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "100 = 5x^2 - 5x\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "subtract 100 from both sides of this equation to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5x^2 - 5x - 100 = 0 which is a quadratic equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "divide both sides of this equation by 5 to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2 - x - 20 = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this factors out to be:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(x-5)*(x+4) = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for x to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = 5 or x = -4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "substitute in original quadratic equation to confirm these answers are good.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5x^2 - 5x = 100 becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "80 + 20 = 100 when x = -4 and becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "125 - 25 = 100 when x = 5.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "both solutions are good.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "plug these solutions into your original equation that you started with to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(x-1) + log(5x) = 2 becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(4) + log(25) = 2 when x = 5.   \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solving this equations gets 2 = 2 confirming x = 5 is a good solution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(x-1) + log(5x) = 2 becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(-5) + log(-20) = 2 when x = -4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this solutions is not valid because you can't take the log of a negative number.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your only valid solution is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = 5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );