document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=244996>Question 244996</A>: <I><SPAN STYLE=\"word-break: break-all;\"> write the expression as a sum or difference of logarithms.<BR>\n" );
document.write( "log(base a)(square root of 16-x^2)<BR>\n" );
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document.write( "so far ive tried log(base a) 16-x^2)^1/2...and then 1/2 log(base a)(16-x^2)...but  i dont thinki can do this 1/2log(base a)16-1/2log(base a)x^2?? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #179114 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=179114\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> equation to solve is:\r<BR>\n" );
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document.write( "log(base a)(square root of 16-x^2)\r<BR>\n" );
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document.write( "this would be written as:\r<BR>\n" );
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document.write( "log(a,(sqrt(16-x^2))) which looks like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C%28sqrt%2816-x%5E2%29%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C%28sqrt%2816-x%5E2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "since sqrt(b) is the same as b^(1/2), this equation can be changed to look like:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C%28%2816-x%5E2%29%5E%281%2F2%29%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C%28%2816-x%5E2%29%5E%281%2F2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "since log(m^n) = n*log(m), this equation can be changed to look like:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29+%2A+log%28a%2C%2816-x%5E2%29%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29+%2A+log%28a%2C%2816-x%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "since 16-x^2 is equal to (4-x)*(4+x), this equation can be changed to look like:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29+%2A+log%28a%2C%28%284-x%29%2A%284%2Bx%29%29%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29+%2A+log%28a%2C%28%284-x%29%2A%284%2Bx%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "since log(m*n) = log(m) + log(n), this equation can be changed to look like:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29+%2A+%28log%28a%2C%284-x%29%29+%2B+log%28a%2C%284%2Bx%29%29%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29+%2A+%28log%28a%2C%284-x%29%29+%2B+log%28a%2C%284%2Bx%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I don't believe it can be transformed any further.\r<BR>\n" );
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document.write( "to confirm this answer is good, then substitute some values for a and x and see if the original equation and the final equation give you the same answer.\r<BR>\n" );
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document.write( "let a = 10 (calculator can do logs to the base 10)<BR>\n" );
document.write( "let x = 3 (random selection of not too big a number and small enough so the square root doesn't go negative)\r<BR>\n" );
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document.write( "original equation is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C%28sqrt%2816-x%5E2%29%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C%28sqrt%2816-x%5E2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which turns out to be equal to .42254902\r<BR>\n" );
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document.write( "final equation is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29+%2A+%28log%28a%2C%284-x%29%29+%2B+log%28a%2C%284%2Bx%29%29%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29+%2A+%28log%28a%2C%284-x%29%29+%2B+log%28a%2C%284%2Bx%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which turns out to be .42254902\r<BR>\n" );
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document.write( "these answers are the same so it looks like the transformed equation is accurate.\r<BR>\n" );
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document.write( "there are 3 basic transformation formulas you need to know.\r<BR>\n" );
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document.write( "log(m^n) = n*log(m)\r<BR>\n" );
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document.write( "log(m*n) = log(m) + log(n)\r<BR>\n" );
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document.write( "log(m/n) = log(m) - log(n)\r<BR>\n" );
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document.write( "you also need to make sure that the multiplier works on all that needs to be multiplied.\r<BR>\n" );
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document.write( "in our case we had the equivalent of:\r<BR>\n" );
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document.write( "log((a*b)^c)\r<BR>\n" );
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document.write( "this became c * log(a*b)\r<BR>\n" );
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document.write( "then this became c * ( log(a) + log(b))\r<BR>\n" );
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document.write( "the c had to multiply them both as I did above with the 1/2 * (... + ...)\r<BR>\n" );
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document.write( "you also need to know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%28n%2Cx%29 ALIGN=MIDDLE  ALT=\"root%28n%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2Fn%29 ALIGN=MIDDLE  ALT=\"x%5E%281%2Fn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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