document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=268185>Question 268185</A>: <I><SPAN STYLE=\"word-break: break-all;\"> express as a single logarithm<BR>\n" );
document.write( "log a 7 + log a 12<BR>\n" );
document.write( "choose from one of these answers<BR>\n" );
document.write( "log a 84<BR>\n" );
document.write( "log 2a 19<BR>\n" );
document.write( "log 2a 84<BR>\n" );
document.write( "log a 19 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #196635 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=196635\">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=log%28a%2C+%287%29%29+%2B+log%28a%2C+%2812%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%287%29%29+%2B+log%28a%2C+%2812%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Logarithms can only be added if the bases and the arguments are the same. Your logarithms have the same base, a, but the arguments are different. So we cannot add them.<br><BR>\n" );
document.write( "So how can <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%287%29%29+%2B+log%28a%2C+%2812%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%287%29%29+%2B+log%28a%2C+%2812%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > be expressed as a single logarithm like all the answers? Well, we have a property of logarithms which allows us to combine two logarithms which have a addition symbol between them: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This property only requires that the bases are the same (and the logarithms have coefficients of 1). Using this on your expression we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%287%2A12%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%287%2A12%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%2884%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%2884%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "When the bases and arguments are the same we can add (or subtract) logarithms. But when we do so, only the coefficients change. After the addition (or subtraction) the bases and arguments of the logarithm <b>are still the same</b>! For example:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%2815%29%29+%2B+3log%28a%2C+%2815%29%29+=+4log%28a%2C+%2815%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%2815%29%29+%2B+3log%28a%2C+%2815%29%29+=+4log%28a%2C+%2815%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It's just like<BR>\n" );
document.write( "x + 3x = 4x.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );