document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=540310>Question 540310</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am given the equation 4^x+4=5^2x+5 and I have to answer it in the form of a log. My professor did not go over something like this so I have no idea on where to start the problem. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #353693 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=353693\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You are given to solve for x:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28x%2B4%29=5%5E%282x%2B5%29 ALIGN=MIDDLE  ALT=\"4%5E%28x%2B4%29=5%5E%282x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Any time you have an unknown in the exponent, one of the things you should think of is using logarithms. Logarithms have a very useful property for such problems. .<BR>\n" );
document.write( "Let's take the logarithm of both sides. We'll use logarithms to the base 10, but we could use natural logs as well. Taking the logarithm of both sides results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C4%5E%28x%2B4%29%29+=+log%2810%2C5%5E%282x%2B5%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C4%5E%28x%2B4%29%29+=+log%2810%2C5%5E%282x%2B5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "A useful property of logarithms is that exponents can be brought out as the multiplier of the logarithm operator. Bringing these two logarithms out as multipliers results in the equation becoming:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+4%29%2Alog%2810%2C4%29+=+%282x+%2B+5%29%2Alog%2810%2C5%29 ALIGN=MIDDLE  ALT=\"%28x+%2B+4%29%2Alog%2810%2C4%29+=+%282x+%2B+5%29%2Alog%2810%2C5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C4%29 ALIGN=MIDDLE  ALT=\"log%2810%2C4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is just a number that you can get from a scientific calculator. Enter 4 and press the \"log\" key. To six decimal places you should see the answer of 0.602056. [This is the exponent you raise the base 10 to that will give you 4.]<BR>\n" );
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document.write( "Similarly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C5%29 ALIGN=MIDDLE  ALT=\"log%2810%2C5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also just a number. Use your calculator to finds its value. You should get (to six decimal places) 0.698970. [This is the exponent you raise the base 10 to that will give you 5.]<BR>\n" );
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document.write( "Substitute these two values into the log equation to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+4%29%2A%280.602056%29+=+%282x+%2B+5%29%2A%280.698970%29 ALIGN=MIDDLE  ALT=\"%28x+%2B+4%29%2A%280.602056%29+=+%282x+%2B+5%29%2A%280.698970%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "From here on out, its a fairly straightforward algebra problem.<BR>\n" );
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document.write( "Do the distributed multiplication on the left side by multiplying 0.602056 times each of the two terms in the parentheses to make the left side become as shown below:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.602056x+%2B+4%2A0.602056+=+%282x+%2B+5%29%2A%280.698970%29 ALIGN=MIDDLE  ALT=\"0.602056x+%2B+4%2A0.602056+=+%282x+%2B+5%29%2A%280.698970%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The 4 times 0.602056 multiplies out to 2.408224 and the equation simplifies to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.602056x+%2B+2.408224+=+%282x+%2B+5%29%2A%280.698970%29 ALIGN=MIDDLE  ALT=\"0.602056x+%2B+2.408224+=+%282x+%2B+5%29%2A%280.698970%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next, follow the same procedure of distributed multiplication on the right side. Multiply 0.698970 times each of the two terms in parentheses to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.602056x+%2B+2.408224+=+1.397940x+%2B+3.494850 ALIGN=MIDDLE  ALT=\"0.602056x+%2B+2.408224+=+1.397940x+%2B+3.494850\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Move the constant from the left side by subtracting 2.408224 from both sides. (On the right side the 3.494850 has 2.408224 subtracted from it.) The equation then is simplified to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.602056x+=+1.397940x+%2B+1.086626 ALIGN=MIDDLE  ALT=\"0.602056x+=+1.397940x+%2B+1.086626\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then transfer the 1.397940x from the right side to the left side by subtracting 1.397940x from both sides. The equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.795884x+=+1.086626 ALIGN=MIDDLE  ALT=\"-0.795884x+=+1.086626\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve for x by dividing both sides by -0.795884 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1.086626%2F-0.705584 ALIGN=MIDDLE  ALT=\"x+=+1.086626%2F-0.705584\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and this equals:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1.365607 ALIGN=MIDDLE  ALT=\"x+=+-1.365607\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Let's try this in the original problem to see if it checks.<BR>\n" );
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document.write( "Start with:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28x%2B4%29=5%5E%282x%2B5%29 ALIGN=MIDDLE  ALT=\"4%5E%28x%2B4%29=5%5E%282x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substitute -1.365607 for x:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28-1.365607%2B4%29+=+5%5E%282%2A-1.365607%2B5%29 ALIGN=MIDDLE  ALT=\"4%5E%28-1.365607%2B4%29+=+5%5E%282%2A-1.365607%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Algebraically simplify the two exponents:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%282.634393%29+=+5%5E%28-2.731214%2B5%29 ALIGN=MIDDLE  ALT=\"4%5E%282.634393%29+=+5%5E%28-2.731214%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The exponent on the right side sums to give:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%282.634393%29+=+5%5E%282.268786%29 ALIGN=MIDDLE  ALT=\"4%5E%282.634393%29+=+5%5E%282.268786%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Use a scientific calculator with an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Ey ALIGN=MIDDLE  ALT=\"x%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" > function key to find the following simplifications to each side of this equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=38.553395=38.531273 ALIGN=MIDDLE  ALT=\"38.553395=38.531273\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Pretty close to equal on both sides. The round off errors of the logarithms cause some differences. And I could have made some fat-fingered keying errors on the calculator also. Check my work. This near equality tells us that the value of x is accurate within a \"reasonable\" amount.<BR>\n" );
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document.write( "[A little closer value of x (by carrying more decimal places on a calculator) is x = -1.36529380395]<BR>\n" );
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document.write( "The process of bringing exponents outside of logarithms as multipliers is correct. Hope you can translate this discussion into an understanding of how you can handle exponents in which a variable appears.<BR>\n" );
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document.write( " </SPAN>\n" );
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