document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1009879>Question 1009879</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the exponential equation by expressing each side as a power of the same base and then equating exponents.   \r<BR>\n" );
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document.write( "(x-8)/(3^8) = sqrt ( 3)\r<BR>\n" );
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document.write( "The solution set is {    } </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #625377 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=625377\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your equation is:\r<BR>\n" );
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document.write( "(x-8) / 3^8 = sqrt(3)\r<BR>\n" );
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document.write( "since sqrt(3) = 3^(1/2), your equation becomes:\r<BR>\n" );
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document.write( "(x-8) / 3^8 = 3^(1/2)\r<BR>\n" );
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document.write( "(x-8) = 3^y if and only if log3(x-8) = y\r<BR>\n" );
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document.write( "this means that (x-8) = 3^(log3(x-8))\r<BR>\n" );
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document.write( "your equation becomes:\r<BR>\n" );
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document.write( "3^(log3(x-8)) / 3^8 = 3^(1/2)\r<BR>\n" );
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document.write( "since 3^(log3(x-8)) / 3^8 is equivalent to 3^(log3(x-8) - 8), your equation becomes:\r<BR>\n" );
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document.write( "3^(log3(x-8)-8) = 3^(1/2)\r<BR>\n" );
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document.write( "this is true if and only if log3(x-8)-8) = 1/2\r<BR>\n" );
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document.write( "add 8 to both sides to get log3(x-8) = 8 + 1/2\r<BR>\n" );
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document.write( "simplify to get log3(x-8) = 17/2\r<BR>\n" );
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document.write( "this is true if and only if 3^(17/2) = x-8\r<BR>\n" );
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document.write( "add 8 to both sides of this equation to get x = 8 + 3^(17/2)\r<BR>\n" );
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document.write( "to confirm the solution is good, go back to the original equation and replace x with (8 + 3^(17/2)\r<BR>\n" );
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document.write( "the original equation is (x-8) / 3^8 = sqrt(3)\r<BR>\n" );
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document.write( "after you replace x, the equation becomes:\r<BR>\n" );
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document.write( "(8 + 3^(17/2) - 8) / 3^8 = 3^(1/2\r<BR>\n" );
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document.write( "the 8 - 8 cancels out and you are left with:\r<BR>\n" );
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document.write( "3^(17/2) / 3^8 = 3^(1/2)\r<BR>\n" );
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document.write( "since x^a / x^b = x^(a-b), this equation becomes:\r<BR>\n" );
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document.write( "3^((17/2)-8) = 3^(1/2)\r<BR>\n" );
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document.write( "make common denominators of the fractions to get:\r<BR>\n" );
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document.write( "3^((17/2)-(16/2)) = 3^(1/2)\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "3^(1/2) = 3^(1/2)\r<BR>\n" );
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document.write( "this confirms the solution is correct.\r<BR>\n" );
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document.write( "the solution is x = 8 + 3^(17/2)\r<BR>\n" );
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document.write( "this confirms the solution is correct.\r<BR>\n" );
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document.write( "another way you could have solved it is as follows:\r<BR>\n" );
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document.write( "start with:\r<BR>\n" );
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document.write( "(x-8)/3^8 = 3^(1/2)\r<BR>\n" );
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document.write( "multiply both sides of this equation by 3^8 to get:\r<BR>\n" );
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document.write( "x-8 = 3^(1/2) * 3^8)\r<BR>\n" );
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document.write( "this becomes x-8 = 3^(17/2)\r<BR>\n" );
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document.write( "add 8 to both sides of this eqution to get:\r<BR>\n" );
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document.write( "x = 3^(17/2) + 8\r<BR>\n" );
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document.write( "this way is much quicker but the problem stated that everything had to be put in base 3 to an exponent form and so there was more to do in order to accomplish that.\r<BR>\n" );
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document.write( "either way you get the same answer which is always a good thing.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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