document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=268905>Question 268905</A>: <I><SPAN STYLE=\"word-break: break-all;\"> write as the sum or difference of two or more logarithms:<BR>\n" );
document.write( "1.log(4x)<BR>\n" );
document.write( "2.ln 15<BR>\n" );
document.write( "3.log 3/xy </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #197388 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=197388\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are many possible answers to these problems. The purpose of these problems, I think, is to get you to show that you know how to use the properties of logarithms. So I will solve the problems this way.<br><BR>\n" );
document.write( "The two properties of logarithms that we will need are:<ul><li><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If the argument of a logarithm is or can written as a product (multiplication), then this property let's you rewrite it as the sum of the logarithms of the factors.</li><li><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If the argument of a logarithm is or can written as a quotient (fraction or division), then this property let's you rewrite it as the difference of the logarithms of the numerator and denominator (divident and divisor).</li></ul><BR>\n" );
document.write( "They allow us to take a single logarithm and rewrite it as the sum or difference of logarithms. Let's see how they work on your problems.<br><BR>\n" );
document.write( "1. log(4x)<BR>\n" );
document.write( "There are many ways to express 4x as a product/multiplication. So there are many ways to rewrite log(4x). Here are a few:<BR>\n" );
document.write( "log(4x) = log(4*x) = log(4) + log(x)<BR>\n" );
document.write( "log(4x) = log(2*2x) = log(2) + log(2x)<BR>\n" );
document.write( "log(4x) = log(16x*(1/4) = log(16x) + log(1/4)<BR>\n" );
document.write( "We could also write 4x as a quotient and use the second property. For example:<BR>\n" );
document.write( "log(4x) = log(20x/5) = log(20x) - log(5)<br><BR>\n" );
document.write( "2. ln(15)<BR>\n" );
document.write( "A product we could use for 15 could be 3*5 so<BR>\n" );
document.write( "ln(15) = ln(3*5) = ln(3) + ln(5)<BR>\n" );
document.write( "A quotient we could use for 15 could be 45/3 do<BR>\n" );
document.write( "ln(15) = ln(45/3) = ln(45) - ln(3)<br><BR>\n" );
document.write( "3. log(3/xy)<BR>\n" );
document.write( "Again, there are many many ways we could do this. Probably the most obvious is to start with the second property:<BR>\n" );
document.write( "log(3/xy) = log(3) - log(xy)<BR>\n" );
document.write( "You might be able to stop here. But, since the argument of the second logarithm is a product, we can use the first property on it. (Note the use of parentheses. They are especially important when replacing one logarithm with an expression of two logarithms!)<BR>\n" );
document.write( "log(3/xy) = log(3) - (log(x) + log(y))}}}<BR>\n" );
document.write( "And because of the \"-\" in front of the parentheses, we should subtract both of the terms inside:<BR>\n" );
document.write( "log(3/xy) = log(3) - log(x) - log(y)}}}<BR>\n" );
document.write( "(Without the use of parentheses, it would be very easy to end up with a \"+\" in front of log(y)!) </SPAN>\n" );
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