document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=553685>Question 553685</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 5^(3y-4)=685<BR>\n" );
document.write( "y=? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #361018 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=361018\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You were asked to solve for y in the equation:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E%283y-4%29=+685 ALIGN=MIDDLE  ALT=\"5%5E%283y-4%29=+685\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Whenever you see an unknown in the exponent, you should think that taking a logarithm of the term is probably the way to solve for the unknown.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "This being the case, let's take the logarithm of the term on the left side. But whatever we do to the left side, we must also do to the right side in order to maintain the equality. So we'll take the log of both sides. What base should we use for the logs? Doesn't make any difference as long as we use the same based logs for both sides of the equation. However, from a practical standpoint (because we can use a calculator to easily find these logarithms) we should use either natural logarithms or base 10 logarithms. Let's use base 10. When we take the log of both sides we get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%285%5E%283y-4%29%29+=+log%28685%29 ALIGN=MIDDLE  ALT=\"log%285%5E%283y-4%29%29+=+log%28685%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "A general property of logarithms is that when you take the logarithm of a quantity that has an exponent, you can bring the exponent out as a multiplier of the logarithm. In this problem, on the left side you can bring the (3y-4) out to multiply the log and then equation then becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283y-4%29log%285%29+=+log%28685%29 ALIGN=MIDDLE  ALT=\"%283y-4%29log%285%29+=+log%28685%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now divide both the left and right sides by log(5). On the left side you are left with just 3y - 4 and the equation is then:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3y+-+4+=+log%28685%29%2Flog%285%29 ALIGN=MIDDLE  ALT=\"3y+-+4+=+log%28685%29%2Flog%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "On the right side the log(685) is just a number we can find with a calculator, and the same thing can be said for log(5). (Remember that we are working in base 10 logarithms.) Enter 685 on your calculator and press the log key to find that log(685) = 2.835056102. Then enter 5 and press the log key to get that log(5) = 0.698970004. By substituting these values the equation becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3y+-+4+=+2.835690571%2F0.698970004 ALIGN=MIDDLE  ALT=\"3y+-+4+=+2.835690571%2F0.698970004\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Doing the division on the right side reduces the equation to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3y-4+=+4.056956027 ALIGN=MIDDLE  ALT=\"3y-4+=+4.056956027\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Then get rid of the -4 on the left side by adding +4 to both sides to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3y+=+8.056956027 ALIGN=MIDDLE  ALT=\"3y+=+8.056956027\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Finally divide both sides by 3 to get the answer:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2.685652009 ALIGN=MIDDLE  ALT=\"y+=+2.685652009\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope this helps you to understand how logarithms can help you to solve for unknowns when they appear in an exponent.<BR>\n" );
document.write( ".<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );