document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=83735>Question 83735</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the equation:\r<BR>\n" );
document.write( "\n" );
document.write( "(1/6)^x = 1/216 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #60227 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=60227\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You can do this problem without using logarithms if you wish.  The process for doing it <BR>\n" );
document.write( "involves:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "First: use the power rule of exponents to raise <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F6 ALIGN=MIDDLE  ALT=\"1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the x power.  By this  rule the<BR>\n" );
document.write( "exponent \"x\" is applied to both the numerator and the denominator so the given equation becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5Ex%2F6%5Ex+=+1%2F216 ALIGN=MIDDLE  ALT=\"1%5Ex%2F6%5Ex+=+1%2F216\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "But 1 raised to the \"x\" power is always 1 no matter what the value of \"x\" is. So the equation <BR>\n" );
document.write( "simplifies to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F6%5Ex+=+1%2F216 ALIGN=MIDDLE  ALT=\"1%2F6%5Ex+=+1%2F216\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Notice that the numerators are equal ... both of them are 1.  So for this equation to be<BR>\n" );
document.write( "true, the denominators must also be equal.  Therefore you can say that:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%5Ex+=+216 ALIGN=MIDDLE  ALT=\"6%5Ex+=+216\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Six squared is 36 and if you multiply that by 6 you get 216.  So you can say that <BR>\n" );
document.write( "6 cubed = 216 so \"x\" must be 3 to indicate that 6 is cubed.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "If you have to do the problem by using logarithms, then start with the given equation:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F6%29%5Ex+=+1%2F216 ALIGN=MIDDLE  ALT=\"%281%2F6%29%5Ex+=+1%2F216\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "and take the logarithm of both sides:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%281%2F6%29%5Ex%29+=+log%281%2F216%29 ALIGN=MIDDLE  ALT=\"log%28%281%2F6%29%5Ex%29+=+log%281%2F216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now apply two rules of logarithms.  The first rule is that in a logarithm of a quantity<BR>\n" );
document.write( "that is raised to an exponent, the exponent can be brought out to become the multiplier<BR>\n" );
document.write( "of the logarithm.  Applying that rule to the left side of the equation, makes the equation<BR>\n" );
document.write( "become:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28log%281%2F6%29%29+=+log%281%2F216%29 ALIGN=MIDDLE  ALT=\"x%2A%28log%281%2F6%29%29+=+log%281%2F216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "The second rule of logarithms to apply is the one that says the log of a fraction is<BR>\n" );
document.write( "equal to the log of the numerator minus the log of the denominator.  Applying that rule<BR>\n" );
document.write( "to both sides in this problem gives:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28log%281%29-log%286%29%29+=+log%281%29-+log+%28216%29 ALIGN=MIDDLE  ALT=\"x%2A%28log%281%29-log%286%29%29+=+log%281%29-+log+%28216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now use the fact that log(1) = 0 and substitute 0 for log(1) on both sides to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%280-log%286%29%29+=+0+-+log+216 ALIGN=MIDDLE  ALT=\"x%2A%280-log%286%29%29+=+0+-+log+216\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "and this simplifies to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28-log%286%29%29+=+-log%28216%29 ALIGN=MIDDLE  ALT=\"x%2A%28-log%286%29%29+=+-log%28216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "If you multiply both sides of this equation by -1 you get rid of the minus signs and the<BR>\n" );
document.write( "equation becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Alog%286%29+=+log%28216%29 ALIGN=MIDDLE  ALT=\"x%2Alog%286%29+=+log%28216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Divide both sides of this equation by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%286%29 ALIGN=MIDDLE  ALT=\"log%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve for x.  This division <BR>\n" );
document.write( "leads to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+log%28216%29%2Flog%286%29 ALIGN=MIDDLE  ALT=\"x+=+log%28216%29%2Flog%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now use your calculator to find that for base 10 logs <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28216%29+=+2.334453751 ALIGN=MIDDLE  ALT=\"log%28216%29+=+2.334453751\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%286%29+=+0.77815125 ALIGN=MIDDLE  ALT=\"log%286%29+=+0.77815125\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Substitute these two values to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2.334453751%2F0.77815121 ALIGN=MIDDLE  ALT=\"x+=+2.334453751%2F0.77815121\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "And when you do this division on a calculator you get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3 ALIGN=MIDDLE  ALT=\"x+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope this helps you to understand a little more about both logarithms and also about handling<BR>\n" );
document.write( "exponents. <BR>\n" );
document.write( ". </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );