document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=203386>Question 203386</A>: <I><SPAN STYLE=\"word-break: break-all;\"> use logarithms to solve each equation:\r<BR>\n" );
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document.write( "3(superscript x)= 5(superscript 5x-1) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #153450 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=153450\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 3 (superscript x) is usually referred to as 3^x.<BR>\n" );
document.write( "similarly 5 (superscript 5x-1) is usually referred to as 5^(5x-1).<BR>\n" );
document.write( "^ is shift 6 on your keyboard.<BR>\n" );
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document.write( "they should look like the following if I understood you correctly.\r<BR>\n" );
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document.write( "as a general rule, if a = b, then log(a) = log(b), so if we let:<BR>\n" );
document.write( "a = 3^x, and<BR>\n" );
document.write( "b = 5^(5x-1), we get:<BR>\n" );
document.write( "log(3^x) = log(5^(5x-1))<BR>\n" );
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document.write( "by the rules of logarithms:<BR>\n" );
document.write( "since we know, that log(3^x) is the same as x*log(3),<BR>\n" );
document.write( "and we know that log(5^(5x-1)) is the same as (5x-1)*log(5), we get:<BR>\n" );
document.write( "x*log(3) = (5x-1)*log(5)<BR>\n" );
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document.write( "if we remove parentheses from the right side of the equation, we get:<BR>\n" );
document.write( "x*log(3) = 5x*log(5) - log(5)<BR>\n" );
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document.write( "if we add log(5) to both sides of this equation, and we subtract x*log(3) from both sides of this equation, we get:<BR>\n" );
document.write( "log(5) = 5x*log(5) - x*log(3)<BR>\n" );
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document.write( "if we separate the common factor x from the right side of the equation, we get:<BR>\n" );
document.write( "log(5) = x*(5*log(5) - log(3))<BR>\n" );
document.write( "which looks like this:\r<BR>\n" );
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document.write( "if we divide both sides of the equation by (5*log(5) - log(3)), we get:<BR>\n" );
document.write( "x = log(5) / (5*log(5) - log(3))<BR>\n" );
document.write( "which looks like this:\r<BR>\n" );
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document.write( "now it's just a matter of finding log(5) and log(3) on the calculator and substituting in the equation to solve for x.<BR>\n" );
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document.write( "since log(5) = .698970004<BR>\n" );
document.write( "and log(3) = .477121255<BR>\n" );
document.write( "we get:<BR>\n" );
document.write( "x = .698970004 / (5*(.698970004) - .477121255)<BR>\n" );
document.write( "which becomes:<BR>\n" );
document.write( "x = .698970004 / 3.017728767<BR>\n" );
document.write( "which becomes:<BR>\n" );
document.write( "x = .231621215<BR>\n" );
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document.write( "3^x = 3^(.231621215) = 1.289767428<BR>\n" );
document.write( "and<BR>\n" );
document.write( "5^(5x-1) = 5^(5*(.231621215) - 1)<BR>\n" );
document.write( "which becomes:<BR>\n" );
document.write( "5^(5x-1) = 5^(.158106076)<BR>\n" );
document.write( "which becomes:<BR>\n" );
document.write( "5^(5x-1) = 1.289767428<BR>\n" );
document.write( "which means that:<BR>\n" );
document.write( "3^x = 5^(5x-1)<BR>\n" );
document.write( "because they both = 1.289767428<BR>\n" );
document.write( "which means that the value of x is good, and your answer is:<BR>\n" );
document.write( "x = .231621215<BR>\n" );
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