document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209957>Question 1209957</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all x such that<BR>\n" );
document.write( "(2x)^{\log_{10} 2} = (9x)*{\log_{10} 9} + (5x)^{\log_{10} 9} + \log_x 243 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851224 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851224\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the given equation be<BR>\n" );
document.write( "$$(2x)^{\log_{10} 2} = (9x)^{\log_{10} 5} + (5x)^{\log_{10} 9} + \log_x 243$$<BR>\n" );
document.write( "Let's analyze the terms separately.<BR>\n" );
document.write( "First, we have $(2x)^{\log_{10} 2}$. Using the property $a^{b+c} = a^b a^c$, we get<BR>\n" );
document.write( "$$(2x)^{\log_{10} 2} = 2^{\log_{10} 2} x^{\log_{10} 2}$$\r<BR>\n" );
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document.write( "Next, we have $(9x)^{\log_{10} 5}$. Using the same property,<BR>\n" );
document.write( "$$(9x)^{\log_{10} 5} = 9^{\log_{10} 5} x^{\log_{10} 5}$$\r<BR>\n" );
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document.write( "Next, we have $(5x)^{\log_{10} 9}$.<BR>\n" );
document.write( "$$(5x)^{\log_{10} 9} = 5^{\log_{10} 9} x^{\log_{10} 9}$$\r<BR>\n" );
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document.write( "Finally, we have $\log_x 243$. Since $243 = 3^5$, we have $\log_x 243 = \log_x 3^5 = 5 \log_x 3$.\r<BR>\n" );
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document.write( "So the equation is<BR>\n" );
document.write( "$$2^{\log_{10} 2} x^{\log_{10} 2} = 9^{\log_{10} 5} x^{\log_{10} 5} + 5^{\log_{10} 9} x^{\log_{10} 9} + 5 \log_x 3$$\r<BR>\n" );
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document.write( "Let's look at the terms involving $x$.<BR>\n" );
document.write( "$x^{\log_{10} 2}$, $x^{\log_{10} 5}$, $x^{\log_{10} 9}$, and $\log_x 3$.\r<BR>\n" );
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document.write( "We can use the change of base formula for logarithms: $\log_x 3 = \frac{\log_{10} 3}{\log_{10} x}$.\r<BR>\n" );
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document.write( "So the equation becomes<BR>\n" );
document.write( "$$2^{\log_{10} 2} x^{\log_{10} 2} = 9^{\log_{10} 5} x^{\log_{10} 5} + 5^{\log_{10} 9} x^{\log_{10} 9} + 5 \frac{\log_{10} 3}{\log_{10} x}$$\r<BR>\n" );
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document.write( "Let's try $x=10$.<BR>\n" );
document.write( "Then $\log_{10} 2$, $\log_{10} 5$, $\log_{10} 9$ are all constants. Also $\log_{10} x = \log_{10} 10 = 1$.<BR>\n" );
document.write( "$\log_x 3 = \log_{10} 3$.\r<BR>\n" );
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document.write( "The equation becomes<BR>\n" );
document.write( "$$2^{\log_{10} 2} 10^{\log_{10} 2} = 9^{\log_{10} 5} 10^{\log_{10} 5} + 5^{\log_{10} 9} 10^{\log_{10} 9} + 5 \log_{10} 3$$\r<BR>\n" );
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document.write( "$10^{\log_{10} 2} = 2$, $10^{\log_{10} 5} = 5$, $10^{\log_{10} 9} = 9$.\r<BR>\n" );
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document.write( "$$2^{\log_{10} 2} \cdot 2 = 9^{\log_{10} 5} \cdot 5 + 5^{\log_{10} 9} \cdot 9 + 5 \log_{10} 3$$\r<BR>\n" );
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document.write( "$$2^{1 + \log_{10} 2} = 5 \cdot 9^{\log_{10} 5} + 9 \cdot 5^{\log_{10} 9} + 5 \log_{10} 3$$\r<BR>\n" );
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document.write( "Let's try $x=3$.<BR>\n" );
document.write( "$$(2\cdot3)^{\log_{10} 2} = (9\cdot3)^{\log_{10} 5} + (5\cdot3)^{\log_{10} 9} + \log_3 243$$<BR>\n" );
document.write( "$$6^{\log_{10} 2} = 27^{\log_{10} 5} + 15^{\log_{10} 9} + 5$$\r<BR>\n" );
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document.write( "This doesn't seem to have an easy solution.\r<BR>\n" );
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document.write( "Let's check if $x=10$ works.<BR>\n" );
document.write( "$(20)^{\log_{10}2} = (90)^{\log_{10}5} + (50)^{\log_{10}9} + \log_{10}243$<BR>\n" );
document.write( "$2^{\log_{10}2} 10^{\log_{10}2} = 9^{\log_{10}5} 10^{\log_{10}5} + 5^{\log_{10}9} 10^{\log_{10}9} + \log_{10}243$<BR>\n" );
document.write( "$2^{\log_{10}2} \cdot 2 = 9^{\log_{10}5} \cdot 5 + 5^{\log_{10}9} \cdot 9 + 5\log_{10}3$<BR>\n" );
document.write( "$2^{1+\log_{10}2} = 5 \cdot 9^{\log_{10}5} + 9 \cdot 5^{\log_{10}9} + 5\log_{10}3$\r<BR>\n" );
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document.write( "This equation is very difficult to solve analytically.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{10}$<BR>\n" );
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