document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209355>Question 1209355</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find x if \log_2 (\log_3 x) = \log_4 x. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848630 by </B><A HREF=/tutors/aboutme.mpl?userid=yurtman><B>yurtman(42)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=yurtman><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848630\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Change Bases**\r<BR>\n" );
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document.write( "* **Use the change-of-base formula:**<BR>\n" );
document.write( "    * log<sub>a</sub>(b) = log<sub>c</sub>(b) / log<sub>c</sub>(a) \r<BR>\n" );
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document.write( "* Apply this to both sides of the equation:<BR>\n" );
document.write( "    * log<sub>2</sub>(log<sub>3</sub>(x)) = log<sub>4</sub>(x) <BR>\n" );
document.write( "    * log<sub>2</sub>(log<sub>3</sub>(x)) = log<sub>2</sub>(x) / log<sub>2</sub>(4)<BR>\n" );
document.write( "    * log<sub>2</sub>(log<sub>3</sub>(x)) = log<sub>2</sub>(x) / 2 \r<BR>\n" );
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document.write( "**2. Simplify**\r<BR>\n" );
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document.write( "* Multiply both sides by 2:<BR>\n" );
document.write( "    * 2 * log<sub>2</sub>(log<sub>3</sub>(x)) = log<sub>2</sub>(x)\r<BR>\n" );
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document.write( "* Apply the power rule of logarithms:<BR>\n" );
document.write( "    * log<sub>2</sub>((log<sub>3</sub>(x))²) = log<sub>2</sub>(x)\r<BR>\n" );
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document.write( "**3. Equate Arguments**\r<BR>\n" );
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document.write( "* Since the bases of the logarithms are the same (base 2), we can equate the arguments:<BR>\n" );
document.write( "    * (log<sub>3</sub>(x))² = x\r<BR>\n" );
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document.write( "**4. Solve for x**\r<BR>\n" );
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document.write( "* This equation is difficult to solve algebraically. <BR>\n" );
document.write( "* **Use numerical methods (like graphing or using a solver function on a calculator) to find the solutions.** \r<BR>\n" );
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document.write( "* **Solutions:**<BR>\n" );
document.write( "    * x ≈ 1 <BR>\n" );
document.write( "    * x ≈ 8.51\r<BR>\n" );
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document.write( "**Therefore, the possible values of x that satisfy the equation log<sub>2</sub>(log<sub>3</sub>(x)) = log<sub>4</sub>(x) are approximately 1 and 8.51.**<BR>\n" );
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