document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=378815>Question 378815</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi I am studting logarithms today with an exam tommorow.Unfortunatly i missed the logarithms lecture as i was sick so I am finding it hard to to solve some of the problems.would really appreciate some help solving these step by step so i can understand it better.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%284x%29=9 ALIGN=MIDDLE  ALT=\"e%5E%284x%29=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E%282x%2B1%29+=+7%5E%28x-1%29 ALIGN=MIDDLE  ALT=\"3%5E%282x%2B1%29+=+7%5E%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "i know iv put up 3 so this might be too muych so if you can only help me solve 1 i would appreciate that too.thanks in advance. Kenneth from Ireland :) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #269074 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=269074\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5Ex+=+125%5E2%5E%28x-1%29 ALIGN=MIDDLE  ALT=\"5%5Ex+=+125%5E2%5E%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Is this really the equation? Or is it:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5Ex+=+125%5E%282%28x-1%29%29 ALIGN=MIDDLE  ALT=\"5%5Ex+=+125%5E%282%28x-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5Ex+=+%28125%5E2%29%5E%28x-1%29 ALIGN=MIDDLE  ALT=\"5%5Ex+=+%28125%5E2%29%5E%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The first equation is extremely difficult. The other two (which are equivalent to each other) are more like the problems given in Math classes. I am going to assume that the last one is correct.<br><BR>\n" );
document.write( "Usually, when the variable is in an exponent, logarithms are used to solve the equation. The exception is when both sides of the equation can be expressed as powers of the same number.<br><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125+=+5%5E3 ALIGN=MIDDLE  ALT=\"125+=+5%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > this equation is one of those that can be solved without logarithms. Substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E3 ALIGN=MIDDLE  ALT=\"5%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > for 125 we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5Ex+=+%28%285%5E3%29%5E2%29%5E%28x-1%29 ALIGN=MIDDLE  ALT=\"5%5Ex+=+%28%285%5E3%29%5E2%29%5E%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The rule for exponents when raising a power to a power is to multiply the exponents. Using this rule (twice) to simplify the right side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5Ex+=+%285%5E6%29%5E%28x-1%29 ALIGN=MIDDLE  ALT=\"5%5Ex+=+%285%5E6%29%5E%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5Ex+=+5%5E%286x-6%29 ALIGN=MIDDLE  ALT=\"5%5Ex+=+5%5E%286x-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Both sides of the equation are now powers of 5. In order for these powers of 5 to be equal, the exponents must be equal. So:<BR>\n" );
document.write( "x = 6x-6<BR>\n" );
document.write( "This is an easy equation to solve. Subtracting 6x from each side:<BR>\n" );
document.write( "-5x = -6<BR>\n" );
document.write( "Dividing both sides by -5 we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+6%2F5 ALIGN=MIDDLE  ALT=\"x+=+6%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E%282x%2B1%29+=+7%5E%28x-1%29 ALIGN=MIDDLE  ALT=\"3%5E%282x%2B1%29+=+7%5E%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In this equation we cannot express both sides as powers of the same number. So we will need to use logarithms. Logarithms of any base can be used. However, there are two things to consider when selecting a base:<ul><li>Using a base for the logarithm that matches the base of one of the exponents will lead to a simpler expression for an answer.</li><li>Using a base for the logarithm that your calculator \"knows\", like base 10 or base e (aka ln), will make it easier to find a decimal approximation of the answer.</li></ul><BR>\n" );
document.write( "So using base 3 or base 7 logarithms will make our answer simpler. But using base 10 or base e logarithms will make finding a decimal approximation easier. I am going to choose base 7:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%287%2C+%283%5E%282x%2B1%29%29%29+=+log%287%2C+%287%5E%28x-1%29%29%29 ALIGN=MIDDLE  ALT=\"log%287%2C+%283%5E%282x%2B1%29%29%29+=+log%287%2C+%287%5E%28x-1%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we use a property of logarithms, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to move the exponent of the argument out in front of the logarithm. It is this property which is the very reason we use logarithms on these equations. It allows us to move the exponent, where the variable is, to a place where we can use \"regular\" Algebra to solve it. Using this property on both sides we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B1%29%2Alog%287%2C+%283%29%29+=+%28x-1%29%2Alog%287%2C+%287%29%29 ALIGN=MIDDLE  ALT=\"%282x%2B1%29%2Alog%287%2C+%283%29%29+=+%28x-1%29%2Alog%287%2C+%287%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "By definition, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%287%2C+%287%29%29+=+1 ALIGN=MIDDLE  ALT=\"log%287%2C+%287%29%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > so this becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B1%29%2Alog%287%2C+%283%29%29+=+x-1 ALIGN=MIDDLE  ALT=\"%282x%2B1%29%2Alog%287%2C+%283%29%29+=+x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "On the left side we will use the Distributive Property to multiply:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2Alog%287%2C+%283%29%29+%2B+log%287%2C+%283%29%29+=+x-1 ALIGN=MIDDLE  ALT=\"2x%2Alog%287%2C+%283%29%29+%2B+log%287%2C+%283%29%29+=+x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Gathering the x terms on one side and the other terms on the other side (by subtracting x and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%287%2C+%283%29%29 ALIGN=MIDDLE  ALT=\"log%287%2C+%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > from each side) we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2Alog%287%2C+%283%29%29+-+x+=+-1+-+log%287%2C+%283%29%29 ALIGN=MIDDLE  ALT=\"2x%2Alog%287%2C+%283%29%29+-+x+=+-1+-+log%287%2C+%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Factoring out x we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%282%2Alog%287%2C+%283%29%29+-+1%29+=+-1+-+log%287%2C+%283%29%29 ALIGN=MIDDLE  ALT=\"x%2A%282%2Alog%287%2C+%283%29%29+-+1%29+=+-1+-+log%287%2C+%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And dividing both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Alog%287%2C+%283%29%29+-+1%29 ALIGN=MIDDLE  ALT=\"%282%2Alog%287%2C+%283%29%29+-+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1+-+log%287%2C+%283%29%29%29%2F%282%2Alog%287%2C+%283%29%29+-+1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1+-+log%287%2C+%283%29%29%29%2F%282%2Alog%287%2C+%283%29%29+-+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is an exact expression for the solution to the equation. If you want a decimal approximation it is not too late. We can use the base conversion formula, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to convert the base 7 logarithms into either base 10 or base e logarithms.<br><BR>\n" );
document.write( "For the most part I will leave<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%284x%29+=+9 ALIGN=MIDDLE  ALT=\"e%5E%284x%29+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "for you to solve. One hint: There is a clear choice for the base of the logarithm to use. Base e logarithms will make the exact answer simpler (since it matches the base of the only exponential term) <b>and</b> it is a base your calculator \"knows\" so a decimal approximation of the answer will be easy.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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