document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=373280>Question 373280</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1/2 log base a of (x+2) + 1/2 log base a of (x-2) = 2/3 log base a of 27 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #265879 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=265879\">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=%281%2F2%29log%28a%2C+%28x%2B2%29%29+%2B+%281%2F2%29log%28a%2C+%28x-2%29%29+=+%282%2F3%29log%28a%2C+%2827%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "With equations with the variable in one or more arguments (or bases) of logarithms, you often start by transforming the equation into one of the following forms:<BR>\n" );
document.write( "log(expression) = other-expression<BR>\n" );
document.write( "or<BR>\n" );
document.write( "log(expression) = log(other-expression)<br><BR>\n" );
document.write( "Since your equation has no \"non-log\" terms, we will aim for the second form. The coefficients of the logarithms in the second form are 1's. Your equation's logarithms have other coefficients. Fortunately we have a property of logarithms, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%2Alog%28w%2C+%28p%29%29+=+log%28w%2C+%28p%5Eq%29%29 ALIGN=MIDDLE  ALT=\"q%2Alog%28w%2C+%28p%29%29+=+log%28w%2C+%28p%5Eq%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which allows us to \"move\" a coefficient of a logarithm into the argument as an exponent. Using this on your three logarithms we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28%28x%2B2%29%5E%281%2F2%29%29%29+%2B+log%28a%2C+%28%28x-2%29%5E%281%2F2%29%29%29+=+log%28a%2C+%2827%5E%282%2F3%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "An exponent of 1/2 means square root. So I am going to replace the arguments which have this exponent with square roots:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28sqrt%28x%2B2%29%29%29+%2B+log%28a%2C+%28sqrt%28x-2%29%29%29+=+log%28a%2C+%2827%5E%282%2F3%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Also, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27%5E%282%2F3%29 ALIGN=MIDDLE  ALT=\"27%5E%282%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > will simplify<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27%5E%282%2F3%29 ALIGN=MIDDLE  ALT=\"27%5E%282%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27%5E%28%281%2F3%29%2A2%29 ALIGN=MIDDLE  ALT=\"27%5E%28%281%2F3%29%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2827%5E%28%281%2F3%29%29%5E2%29 ALIGN=MIDDLE  ALT=\"%2827%5E%28%281%2F3%29%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28root%283%2C+27%29%5E2%29 ALIGN=MIDDLE  ALT=\"%28root%283%2C+27%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E3+=+27 ALIGN=MIDDLE  ALT=\"3%5E3+=+27\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C+27%29+=+3 ALIGN=MIDDLE  ALT=\"root%283%2C+27%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E2 ALIGN=MIDDLE  ALT=\"3%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "9<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27%5E%282%2F3%29+=+9 ALIGN=MIDDLE  ALT=\"27%5E%282%2F3%29+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This makes our equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28sqrt%28x%2B2%29%29%29+%2B+log%28a%2C+%28sqrt%28x-2%29%29%29+=+log%28a%2C+%289%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The coefficients are now the 1's we wanted. Next we need to combine the two logarithms into one. And we have a property of logarithms, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28w%2C+%28p%29%29+%2B+log%28w%2C+%28q%29%29+=+log%28w%2C+%28p%2Aq%29%29 ALIGN=MIDDLE  ALT=\"log%28w%2C+%28p%29%29+%2B+log%28w%2C+%28q%29%29+=+log%28w%2C+%28p%2Aq%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which allows us to do just that! Using this property on the left side we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28sqrt%28x%2B2%29%2Asqrt%28x-2%29%29%29+=+log%28a%2C+%289%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28sqrt%28x%2B2%29%2Asqrt%28x-2%29%29%29+=+log%28a%2C+%289%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which, using a property of radicals, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%28w%2C+p%29%2Aroot%28w%2C+q%29+=+root%28w%2C+p%2Aq%29 ALIGN=MIDDLE  ALT=\"root%28w%2C+p%29%2Aroot%28w%2C+q%29+=+root%28w%2C+p%2Aq%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28sqrt%28%28x%2B2%29%2A%28x-2%29%29%29%29+=+log%28a%2C+%289%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28sqrt%28%28x%2B2%29%2A%28x-2%29%29%29%29+=+log%28a%2C+%289%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We finally have achieved the second form! From here we use some basic logic:<BR>\n" );
document.write( "If the base a logarithms of two expressions are equal then the two expressions must be equal. So:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28x%2B2%29%2A%28x-2%29%29+=+9 ALIGN=MIDDLE  ALT=\"sqrt%28%28x%2B2%29%2A%28x-2%29%29+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To solve this we need to eliminate the square root. So we square both sides:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28%28x%2B2%29%2A%28x-2%29%29%29%5E2+=+%289%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28%28x%2B2%29%2A%28x-2%29%29%29%5E2+=+%289%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies as follows:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%2A%28x-2%29+=+81 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%2A%28x-2%29+=+81\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+4+=+81 ALIGN=MIDDLE  ALT=\"x%5E2+-+4+=+81\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Adding 4 to each side we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+85 ALIGN=MIDDLE  ALT=\"x%5E2+=+85\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Finding the square root of each side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2%29+=+sqrt%2885%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2%29+=+sqrt%2885%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%29+=+sqrt%2885%29 ALIGN=MIDDLE  ALT=\"abs%28x%29+=+sqrt%2885%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+=+sqrt%2885%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%2885%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-sqrt%2885%29 ALIGN=MIDDLE  ALT=\"x+=+-sqrt%2885%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Whenever you are solving a logarithmic equation you must check your answers (at least to make sure all arguments and bases of logarithms are not zero or negative). And whenever you square both sides of an equation you must check for extraneous solutions. (Extraneous solutions are solutions which work in the squared equation <i>but do not work in the original equation.</i> We have a logarithmic equation <i>and</i> we squared both sides!<br><BR>\n" );
document.write( "When checking, use the original equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29log%28a%2C+%28x%2B2%29%29+%2B+%281%2F2%29log%28a%2C+%28x-2%29%29+=+%282%2F3%29log%28a%2C+%2827%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Checking <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%2885%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%2885%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29log%28a%2C+%28%28sqrt%2885%29%29%2B2%29%29+%2B+%281%2F2%29log%28a%2C+%28%28sqrt%2885%29%29-2%29%29+=+%282%2F3%29log%28a%2C+%2827%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using our properties again:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28%28sqrt%2885%29%2B2%29%5E%281%2F2%29%29%29+%2B+log%28a%2C+%28%28sqrt%2885%29-2%29%5E%281%2F2%29%29%29+=+log%28a%2C+%2827%5E%282%2F3%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28sqrt%28sqrt%2885%29%2B2%29%29%29+%2B+log%28a%2C+%28sqrt%28sqrt%2885%29-2%29%29%29+=+log%28a%2C+%289%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28sqrt%28sqrt%2885%29%2B2%29%2Asqrt%28sqrt%2885%29-2%29%29%29+=+log%28a%2C+%289%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28sqrt%28%28sqrt%2885%29%2B2%29%2A%28sqrt%2885%29-2%29%29%29%29+=+log%28a%2C+%289%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28p%2Bq%29%28p-q%29+=+p%5E2+-+q%5E2 ALIGN=MIDDLE  ALT=\"%28p%2Bq%29%28p-q%29+=+p%5E2+-+q%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2885%29%2B2%29%2Asqrt%2885%29-2%29 ALIGN=MIDDLE  ALT=\"sqrt%2885%29%2B2%29%2Asqrt%2885%29-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28sqrt%28%28sqrt%2885%29%29%5E2+-+2%5E2%29%29%29+=+log%28a%2C+%289%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28sqrt%28%28sqrt%2885%29%29%5E2+-+2%5E2%29%29%29+=+log%28a%2C+%289%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28sqrt%2885+-+4%29%29%29+=+log%28a%2C+%289%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28sqrt%2885+-+4%29%29%29+=+log%28a%2C+%289%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28sqrt%2881%29%29%29+=+log%28a%2C+%289%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28sqrt%2881%29%29%29+=+log%28a%2C+%289%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%289%29%29+=+log%28a%2C+%289%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%289%29%29+=+log%28a%2C+%289%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Check!!<br><BR>\n" );
document.write( "Checking <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-sqrt%2885%29 ALIGN=MIDDLE  ALT=\"x+=+-sqrt%2885%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29log%28a%2C+%28%28-sqrt%2885%29%29%2B2%29%29+%2B+%281%2F2%29log%28a%2C+%28%28-sqrt%2885%29%29-2%29%29+=+%282%2F3%29log%28a%2C+%2827%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Already we can see that the two arguments on the left side will be negative (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%2885%29 ALIGN=MIDDLE  ALT=\"-sqrt%2885%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is -9.something). Since arguments of logarithms can never be negative, we must reject this solution!<br><BR>\n" );
document.write( "So the only solution to your equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%2885%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%2885%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );