document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=969711>Question 969711</A>: <I><SPAN STYLE=\"word-break: break-all;\"> May I know how to answer this question? <BR>\n" );
document.write( "Given that log12(3)=p,express each of the following in terms of p<BR>\n" );
document.write( "(i)log sqrt12 (1/9)<BR>\n" );
document.write( "(ii)log1/12(27)<BR>\n" );
document.write( "(iii)log12(1/4)<BR>\n" );
document.write( "(iv) log144(4) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #592497 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=592497\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you are given that:\r<BR>\n" );
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document.write( "log.12(3) = p\r<BR>\n" );
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document.write( "you are asked to find the following:\r<BR>\n" );
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document.write( "number 1:\r<BR>\n" );
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document.write( "find log.sqrt(12)(1/9) in terms of p.\r<BR>\n" );
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document.write( "log.sqrt(12)(1/9) = y if and only if sqrt(12)^y = 1/9\r<BR>\n" );
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document.write( "sqrt(12)^y is the same as 12^(y/2), so the equation becomes:\r<BR>\n" );
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document.write( "12^(y/2) = 1/9\r<BR>\n" );
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document.write( "square both sides of this equation to get 12^y = 1/81.\r<BR>\n" );
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document.write( "this is true if and only if log.12(1/81) = y\r<BR>\n" );
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document.write( "1/81 is the same as 3^-4, so this equation becomes:\r<BR>\n" );
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document.write( "log.12(3^-4) = y\r<BR>\n" );
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document.write( "this is the same as -4*log.12(3) = y\r<BR>\n" );
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document.write( "since log.12(3) = p, this is the same at -4*p = y\r<BR>\n" );
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document.write( "since y = -4*p, then log.sqrt(12)(1/9) = y becomes log.sqrt(12)(1/9) = -4*p.\r<BR>\n" );
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document.write( "that's your solution to the first problem.\r<BR>\n" );
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document.write( "log.sqrt(12)(1/9) = -4*p.\r<BR>\n" );
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document.write( "problem number 2:\r<BR>\n" );
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document.write( "find log.1/12(27) in terms of p.\r<BR>\n" );
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document.write( "set this equal to y and you get log.1/12(27) = y\r<BR>\n" );
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document.write( "this is true if and only if (1/12)^y = 27\r<BR>\n" );
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document.write( "1/12 is the same as 12^-1, so the equation becomes:\r<BR>\n" );
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document.write( "12^-1)^y = 27\r<BR>\n" );
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document.write( "this is the same as 12^-y = 27\r<BR>\n" );
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document.write( "this is the same as 1/12^y = 27\r<BR>\n" );
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document.write( "solve for 12^y to get 12^y = 1/27\r<BR>\n" );
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document.write( "this is true if and only if log.12(1/27) = y\r<BR>\n" );
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document.write( "1/27 is the same as 3^-3, so the equation becomes:\r<BR>\n" );
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document.write( "log.12(3^-3) = y\r<BR>\n" );
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document.write( "this is the same as -3*log.12(3) = y\r<BR>\n" );
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document.write( "log.12(3) = p, so this equation becomes -3*p = y\r<BR>\n" );
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document.write( "since y = -3*p, then the equation of log.1/12(27) = y becomes log.1/12(27) = -3*p.\r<BR>\n" );
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document.write( "that's your solution.\r<BR>\n" );
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document.write( "log.1/12(27) = -3*p\r<BR>\n" );
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document.write( "problem number 3:\r<BR>\n" );
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document.write( "find log.12(1/4) in terms of p.\r<BR>\n" );
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document.write( "set 3^x = 1/4 and solve for x.\r<BR>\n" );
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document.write( "take log of both sides of this equation to get log(3^x) = log(1/4)<BR>\n" );
document.write( "this is the same as x * log(3) = log(1/4).<BR>\n" );
document.write( "solve for x to get x = log(1/4)/log(3).\r<BR>\n" );
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document.write( "your equation of log.12(1/4) is equal to log.12(3^(log(1/4)/log(3)).\r<BR>\n" );
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document.write( "this is the same as log.12(1/4) is equal to log(1/4)/log(3) * log.12(3).\r<BR>\n" );
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document.write( "since log.12(3) is equal to p, then this equation becomes:\r<BR>\n" );
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document.write( "log.12(1/4) is equal to log(1/4)/log(3) * p.\r<BR>\n" );
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document.write( "that's your solution.\r<BR>\n" );
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document.write( "problem 4:\r<BR>\n" );
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document.write( "find  log.144(4) in terms of p.\r<BR>\n" );
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document.write( "log.144(4) = y if and only if 144^y = 4\r<BR>\n" );
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document.write( "this is the same as log.144(4) = y if and only if (12^2)^y = 4 which is the same as 12^(2y) = 4.\r<BR>\n" );
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document.write( "this is true if and only if log.12(4) = 2y\r<BR>\n" );
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document.write( "divide both sides of this equation by 2 to get 1/2 * log.12(4) = y\r<BR>\n" );
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document.write( "this is the same as log.12(4^(1/2) = y which is the same as log.12(2) = y\r<BR>\n" );
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document.write( "set 3^x = 2 and solve for x.\r<BR>\n" );
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document.write( "take the log of both sides of this equation to get log(3^x) = log(2).\r<BR>\n" );
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document.write( "this is the same as x*log(3) = log(2).\r<BR>\n" );
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document.write( "solve for x to get x = log(2)/log(3).\r<BR>\n" );
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document.write( "your equation of log.12(2) = y  becomes log.12(3^(log(2)/log(3)) = y.\r<BR>\n" );
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document.write( "log.12(3^(log(2)/log(3)) = y is the same as log(2)/log(3)*log.12(3) = y.\r<BR>\n" );
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document.write( "since log.12(3) is equal to p, then log(2)/log(3)*log.12(3) = y becomes log(2)/log(3)*p = y.\r<BR>\n" );
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document.write( "since y = log(2)/log(3)*p and since we started with log.144(4) = y, then we get:\r<BR>\n" );
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document.write( "log.144(4) = log(2)/log(3)*p.\r<BR>\n" );
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document.write( "that's your solution.\r<BR>\n" );
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document.write( "to summarize your solutions.\r<BR>\n" );
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document.write( "you are given that log.12(3) = p\r<BR>\n" );
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document.write( "you are asked to find the following in terms of p.\r<BR>\n" );
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document.write( "(i)log sqrt12 (1/9)<BR>\n" );
document.write( "(ii)log1/12(27)<BR>\n" );
document.write( "(iii)log12(1/4)<BR>\n" );
document.write( "(iv) log144(4)\r<BR>\n" );
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document.write( "your solutions are:\r<BR>\n" );
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document.write( "(i)log sqrt12 (1/9) = -4*p<BR>\n" );
document.write( "(ii)log1/12(27) = -3*p<BR>\n" );
document.write( "(iii)log12(1/4) = log(1/4)/log(3) * p<BR>\n" );
document.write( "(iv) log144(4) = log(2)/log(3)*p\r<BR>\n" );
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document.write( "you can confirm all of these using your calculator and the log conversion formula of log.b(x) = log(x)/log(b), where log is the LOG function of your calculator.\r<BR>\n" );
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document.write( "p = log.12(3) which is equal to log(3)/log(12) which is equal to .4421141087\r<BR>\n" );
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document.write( "log.sqrt(12)(1/9) = log(1/9)/log(sqrt(12)) = -1.768456435<BR>\n" );
document.write( "-4*p = -4*.4421141087 = -1.768456435<BR>\n" );
document.write( "they match.\r<BR>\n" );
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document.write( "log.1/12(27) = log(27) / log(1/12) = -1.326342326<BR>\n" );
document.write( "-3*p = -3*.4421141087 = -1.326342326<BR>\n" );
document.write( "they match.\r<BR>\n" );
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document.write( "log.12(1/4) = log(1/4) / log(12) = -.5578858913<BR>\n" );
document.write( "log(1/4) /log(3) * p = -.5578858913<BR>\n" );
document.write( "they match.\r<BR>\n" );
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document.write( "log.144(4) = log(4)/log(144) = .2789429457<BR>\n" );
document.write( "log(2)/log(3)*p = .2789429457<BR>\n" );
document.write( "they match\r<BR>\n" );
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document.write( "all the answers match between the original equations and the final equations so the solutions are good.\r<BR>\n" );
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document.write( "in case you haven't figured it out yet, log.b means log to the base of b.<BR>\n" );
document.write( "similarly log.12 means log to the base of 12.<BR>\n" );
document.write( "similarly log.sqrt(12) means log to the base of sqrt(12).<BR>\n" );
document.write( "similarly log.1/12 means log to the base of 1/12.<BR>\n" );
document.write( "etc.<BR>\n" );
document.write( "i use the . to separate the base from log.<BR>\n" );
document.write( "it makes it a little easiwer to read.<BR>\n" );
document.write( "log1/12 would have been cumbersom.<BR>\n" );
document.write( "log.1/12 makes it easier to see that the base is 1/12.<BR>\n" );
document.write( "this laso avoids the use of log[1/12](x) which is technically more correct but also more difficult to type.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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