document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189461>Question 1189461</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Answer the following logarithmic inequalities.\r<BR>\n" );
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document.write( "1) log3(1-x)>=log3(x+16-x^2)\r<BR>\n" );
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document.write( "(Note: 3 is a base)<BR>\n" );
document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #820838 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=820838\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> in order for this to be true, 1-x must be > 0 and x + 16 - x^2 must be > 0<BR>\n" );
document.write( "this is because the expression within the log sign must be positive.<BR>\n" );
document.write( "you get:<BR>\n" );
document.write( "x < 1 and x > -3.5311 and x < 4.5311.<BR>\n" );
document.write( "since x < 1 is the controlling influence on the x < side of the inequality, you get x > -3.5311 and x < 1<BR>\n" );
document.write( "the admissible range is therefore -3.5311 < x < 1<BR>\n" );
document.write( "within that range, you need to find out when log3(1-x) >= log3(-x^2 + x + 16).<BR>\n" );
document.write( "if you set them equal to each other, you get:<BR>\n" );
document.write( "log3(1-x) = log3(-x^2 + x + 16)<BR>\n" );
document.write( "this is true if and only if 1-x = -x^2 + x + 16<BR>\n" );
document.write( "subtract (1-x) from both sides of this equation to get:<BR>\n" );
document.write( "0 = -x^2 + x + 16 -(1-x)<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "0 = -x^2 + 2x + 15<BR>\n" );
document.write( "multiply both sides of this equation by -1 to get:<BR>\n" );
document.write( "0 = x^2 - 2x - 15]<BR>\n" );
document.write( "factor this equation to get:<BR>\n" );
document.write( "0 = (x-5) * (x+3)<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = 5 or x = -3<BR>\n" );
document.write( "x can't be 5 since the permissible for the log function is x < 1.<BR>\n" );
document.write( "x can be -3 since that is within the permissible range of x.<BR>\n" );
document.write( "you have 1-x = -x^2 + x + 16 when x = -3.<BR>\n" );
document.write( "this means that log3(1-x) = log(-x^2 + x + 16 when x = -3.<BR>\n" );
document.write( "this means that log3(1-x) - log(-x^2 + x + 16 = 0 when x = -3.<BR>\n" );
document.write( "log3(1-x) will be greater than log3(-x^2 + x + 16) when x is < -3 or when x is > -3<BR>\n" );
document.write( "at his point, you just need to evaluate when x is less than -3 and when x is greater than -3.<BR>\n" );
document.write( "you can use the log base conversion formula to test this out using a scientific calculator.<BR>\n" );
document.write( "the log conversion formula is log3(x) = log(x)/log(3)<BR>\n" );
document.write( "i used x = -3.5 for x less than -3 and x = .5 for x greater than -3 because x can't be greater than or equal to 1 and x can't be less than or equal to -3.5311 as we found out earlier.<BR>\n" );
document.write( "when x = -3.5:<BR>\n" );
document.write( "log(1-x) = 1.369070246.<BR>\n" );
document.write( "log(-x^2 + x + 16) = -1.261859507.<BR>\n" );
document.write( "this means that log(1-x)/log(3) is greater than log(-x^2 + x + 16)/log(30 when x is less than -3 and greater than -3.5311<BR>\n" );
document.write( "when x = .5:<BR>\n" );
document.write( "log(1-x) = -.6309297536.<BR>\n" );
document.write( "log(-x^2 + x + 16) = 2.537831533.<BR>\n" );
document.write( "this means that log(1-x)/log(3) is less than log(-x^2 + x + 16)/log(3) when x is greater than -3 and less than x = 1.<BR>\n" );
document.write( "your solution should be that log3(1-x) >= log3(x + 16 - x^2) when x is greater than -3.5311 and x is less than or equal to -3.\r<BR>\n" );
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document.write( "i confess that i didn't figure this out until i graphed the function.<BR>\n" );
document.write( "i used the desmos.com calculator.<BR>\n" );
document.write( "it can be found at <a  rel=nofollow HREF= \"https://www.desmos.com/calculator\" target = \"_blank\">https://www.desmos.com/calculator</a><BR>\n" );
document.write( "the graph looks like this.\r<BR>\n" );
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document.write( "i used a quadratic equation solver to find the 0 points for x + 16 - x^2.<BR>\n" );
document.write( "the 0 point of interest was at x = -3.5311288741493.<BR>\n" );
document.write( "x had to be greater than that for log3(x + 16 - x^2) to be valid.<BR>\n" );
document.write( "consequently, the permissible value of x became:<BR>\n" );
document.write( "x > -3.5311288741493 and x <= -3.\r<BR>\n" );
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document.write( "when x = -3, log3(1-x) = log3(x + 16 - x^2)<BR>\n" );
document.write( "when x < -3 and > -3.5311288741493, log3(1-x) > log3(x + 16 - x^2).\r<BR>\n" );
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document.write( "your solution is that log3(1-x) >= log(x + 16 - x^2) when x > -3.5311288741493 and when x <= -3.\r<BR>\n" );
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document.write( "if you use the graph solution, you can say that x > -3.531 x <= -3.<BR>\n" );
document.write( "this translates to -3.531 < x <= -3.\r<BR>\n" );
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document.write( "tough problem.<BR>\n" );
document.write( "i'm not sure how much rounding they allow, but you can adjust from the unrounded figures as necessary.\r<BR>\n" );
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document.write( "the first thing to do was find the permissible range of log3(1-x) and log3(x + 16 - x^2).<BR>\n" );
document.write( "that told you the that the value of x had to be greater -3.5311.... and less than 1.\r<BR>\n" );
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document.write( "the next thing to do was find out when they were equal.<BR>\n" );
document.write( "that led to x = -3.\r<BR>\n" );
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document.write( "the next thing to do was find out when log3(1-x) was greater than log3(x + 16 - x^2) within the permissible range of x.\r<BR>\n" );
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document.write( "that's when you evaluated in the range of x > -3.5311 and x < -3 and the range of x > -3 and x < 1.\r<BR>\n" );
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document.write( "being able to graph the equation of y = log3(1-x) - log3(x + 16 - x^2) was a big help to visualize where the solution would lie.\r<BR>\n" );
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document.write( "let me know if you have any questions.\r<BR>\n" );
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document.write( "theo\r<BR>\n" );
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