document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186865>Question 1186865</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places. \r<BR>\n" );
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document.write( " 3e^−x+11 = 48\r<BR>\n" );
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document.write( "If you wish to enter log or ln, you must use the keypad\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #817844 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=817844\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i believe this is:\r<BR>\n" );
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document.write( "your equation is, as far as i can tell,\r<BR>\n" );
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document.write( "3e^(-x) + 11 = 48\r<BR>\n" );
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document.write( "if so, subtract 11 from both sides of the equation to get:\r<BR>\n" );
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document.write( "3e^(-x) = 37\r<BR>\n" );
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document.write( "divide both sides of the equation by 3 to get:\r<BR>\n" );
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document.write( "e^(-x) = 37/3\r<BR>\n" );
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document.write( "take thre natural log of both sides of the equation to get:\r<BR>\n" );
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document.write( "ln(e^(-x) = ln(37/3)\r<BR>\n" );
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document.write( "since ln(e^(-x) = -x * ln(e) and, since ln(e) = 1, then ln(e^(-x) = -x.\r<BR>\n" );
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document.write( "your equation becomeds:\r<BR>\n" );
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document.write( "-x = ln(37/3)\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = -ln(37/3) = -2.512305624.\r<BR>\n" );
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document.write( "confirm by replacing x in the original equation to get:\r<BR>\n" );
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document.write( "3e^(-x) + 11 = 48 becomes 3e^(-2.512305624) + 11 = 48\r<BR>\n" );
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document.write( "evaluate to get 48 = 48, confirming the value of x is correct.\r<BR>\n" );
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document.write( "HOWEVER,\r<BR>\n" );
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document.write( "it is also possible that your equation is:\r<BR>\n" );
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document.write( "3e^(−x+11) = 48\r<BR>\n" );
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document.write( "without any parentheses, it's difficult to determine what you meant.\r<BR>\n" );
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document.write( "in this case, divide both sides of the equation by 3 to get:\r<BR>\n" );
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document.write( "e^(-x+11) = 16\r<BR>\n" );
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document.write( "take the natural log of both sides of the equation to get:\r<BR>\n" );
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document.write( "ln(e^(-x+11)) = ln(16)\r<BR>\n" );
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document.write( "since ln(e^(-x+11)) = (-x+11) * ln(e) which is equal to -x+11, your equation becomes:\r<BR>\n" );
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document.write( "-x + 11 = ln(16).\r<BR>\n" );
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document.write( "subtract 11 from both sides of the equation to get:\r<BR>\n" );
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document.write( "-x = ln(16) - 11.\r<BR>\n" );
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document.write( "multiply both sides of the equation by -1 to get:\r<BR>\n" );
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document.write( "x = -ln(16) + 11\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = 8.227411278.\r<BR>\n" );
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document.write( "confirm by replacing x in the original equation to get:\r<BR>\n" );
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document.write( "3e^(−x+11) = 48 becomes 3e^(−8.227411278+11) = 48\r<BR>\n" );
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document.write( "evaluate to get 48 = 48, confirming that the value of x is good.\r<BR>\n" );
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document.write( "if the original equation was 3e^(-x) + 11 = 48, then the solution is x = -2.512305624.\r<BR>\n" );
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document.write( "if the original equation was 3e^(−x+11) = 48, then the solution is x = 8.227411278.\r<BR>\n" );
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document.write( "i would have chosen the first option of 3e^−x+11 = 48 = 3e^(-x) + 11 = 48 because, without the parentheses, the equation becomes:<BR>\n" );
document.write( "3 * e^(-x) + 11 = 48.<BR>\n" );
document.write( "it then becomes:<BR>\n" );
document.write( "(3 * e^(-x) + 11 = 48<BR>\n" );
document.write( "subtract 11 from both sides to get:<BR>\n" );
document.write( "3 * e^(-x) = 37<BR>\n" );
document.write( "divide both sides by 3 to get:<BR>\n" );
document.write( "e^(-x) = 37/3.<BR>\n" );
document.write( "you then took the natural log of both sides to get:<BR>\n" );
document.write( "-x = ln(37/3).<BR>\n" );
document.write( "you then solve for x to get:<BR>\n" );
document.write( "x = -ln(37/3).\r<BR>\n" );
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document.write( "these equations can be graphed.\r<BR>\n" );
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document.write( "the graph is shown below:\r<BR>\n" );
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document.write( "the red equation is y = 3 * e^(-x + 11).\r<BR>\n" );
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document.write( "the blue equation is y = 3 * e^(-x) + 11.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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