document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=83403>Question 83403</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the exponential equation and express approximate solutions to the nearest hundredth.<BR>\n" );
document.write( "7^x + 8 = 38<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 #59909 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=59909\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5Ex+%2B+8+=+38 ALIGN=MIDDLE  ALT=\"7%5Ex+%2B+8+=+38\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "solve for x<BR>\n" );
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document.write( "Isolate the term containing x on the left side and have everything else on the other side.<BR>\n" );
document.write( "You can do this by eliminating +8 from the left side. This is done by subtracting 8 from<BR>\n" );
document.write( "both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5Ex+=+30 ALIGN=MIDDLE  ALT=\"7%5Ex+=+30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now you can use the property of logarithms that says:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%5Eb%29+=+b%2Alog%28a%29 ALIGN=MIDDLE  ALT=\"log%28a%5Eb%29+=+b%2Alog%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Take the logarithm of both sides:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%287%5Ex%29+=+log%2830%29 ALIGN=MIDDLE  ALT=\"log%287%5Ex%29+=+log%2830%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "By using the property given above, we convert the left side of the logarithm equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Alog%287%29+=+log%2830%29 ALIGN=MIDDLE  ALT=\"x%2Alog%287%29+=+log%2830%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now solve for x by dividing both sides by log(7). This division causes the equation <BR>\n" );
document.write( "to become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28log%2830%29%29%2F%28log%287%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28log%2830%29%29%2F%28log%287%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that log(30) and log(7) are just numbers. If these logs are to the base 10 or to the<BR>\n" );
document.write( "base e (base e logs are commonly identified as ln), you can get them from a scientific<BR>\n" );
document.write( "calculator.  Putting 30 into your calculator and pressing the log key will show you that<BR>\n" );
document.write( "in base 10 the log of 30 is 1.477121255. Similarly in base 10 the log of 7 is 0.84509804.<BR>\n" );
document.write( "Substituting these values into the equation for x results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+log%2830%29%2Flog%287%29+=+1.477121255%2F0.84509804+=+1.747869697 ALIGN=MIDDLE  ALT=\"x+=+log%2830%29%2Flog%287%29+=+1.477121255%2F0.84509804+=+1.747869697\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Rounding this answer to the nearest hundredth gives:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1.75 ALIGN=MIDDLE  ALT=\"x+=+1.75\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Let's check to see if this value of x makes the original given equation true. Start by<BR>\n" );
document.write( "substituting 1.75 for x and this makes the left side of original equation become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5E1.75+%2B+8 ALIGN=MIDDLE  ALT=\"7%5E1.75+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "A calculator can be used to find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5E1.75 ALIGN=MIDDLE  ALT=\"7%5E1.75\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30.12461949 ALIGN=MIDDLE  ALT=\"30.12461949\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "When this is substituted for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5E1.75 ALIGN=MIDDLE  ALT=\"7%5E1.75\"   DISABLED_style= \"max-width:90%; height:auto;\" > the left side of the original equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30.12461949+%2B+8+=+38.12461949 ALIGN=MIDDLE  ALT=\"30.12461949+%2B+8+=+38.12461949\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is approximately the 38 that the original problem says it should be.  The difference<BR>\n" );
document.write( "is caused by rounding x to the nearest hundredth instead of carrying it to more decimal <BR>\n" );
document.write( "places.<BR>\n" );
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document.write( "Hope this helps you to understand a property of logarithms that can be used when a variable<BR>\n" );
document.write( "appears as an exponent.<BR>\n" );
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