document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=303567>Question 303567</A>: <I><SPAN STYLE=\"word-break: break-all;\"> log(7)x=5    the answer I got is x=16,954    and also log x =1  the answer I got is x=10       log(4) (x-3) = log(4) (x-3)=1  the answer I got is x=3  just not sure if I am right on these problems...I am having hard time understanding this.....and one I have no clue how to do is  e^t=1000    any help would be greatly appreciated....thanks in advance </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #217564 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=217564\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I presume you are saying the following:\r<BR>\n" );
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document.write( "log of x to the base 7 = 5\r<BR>\n" );
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document.write( "That would be written as log(7,x) = 5\r<BR>\n" );
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document.write( "By the basic definitions of logarithms, that can only be true if:\r<BR>\n" );
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document.write( "7^5 = x\r<BR>\n" );
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document.write( "Since 7^5 = 16807, then your answer has to be that x is equal to 16807.\r<BR>\n" );
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document.write( "Your answer should be (if I interpreted the problem correctly), that:\r<BR>\n" );
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document.write( "log(7,16807) = 5.\r<BR>\n" );
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document.write( "You can use your calculator to confirm that directly by using the log base conversion formula.\r<BR>\n" );
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document.write( "Using that formula:\r<BR>\n" );
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document.write( "log(7,16807) = log(10,16807)/log(10,7)\r<BR>\n" );
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document.write( "Solving that gets you:\r<BR>\n" );
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document.write( "log(7,16807) = 5, confirming that the answer is good.\r<BR>\n" );
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document.write( "Your second problem is log(x) = 1\r<BR>\n" );
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document.write( "Without specifying the base, the base of 10 is implied.\r<BR>\n" );
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document.write( "Your second problem equation becomes log(10,x) = 1\r<BR>\n" );
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document.write( "By the basic definition of logarithms, this is true if and only if:\r<BR>\n" );
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document.write( "10^1 = x\r<BR>\n" );
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document.write( "Since 10^1 = 10, your answer should be that x = 10.\r<BR>\n" );
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document.write( "your equation becomes log(10,10) = 1\r<BR>\n" );
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document.write( "This can written as log(10) = 1\r<BR>\n" );
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document.write( "Use your calculator to get log(10) and you'll see that it is equal to 1.\r<BR>\n" );
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document.write( "Looks like you got that one right.\r<BR>\n" );
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document.write( "Your third problem looks like log(4,(x-3)) = 1.\r<BR>\n" );
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document.write( "By the basic definition of logarithms, this is true if and only if:\r<BR>\n" );
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document.write( "4^1 = x-3 which becomes:\r<BR>\n" );
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document.write( "4 = x-3\r<BR>\n" );
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document.write( "Add 3 to both sides of this equation to get:\r<BR>\n" );
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document.write( "x = 7\r<BR>\n" );
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document.write( "Your answer should be x = 7.\r<BR>\n" );
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document.write( "Your equation becomes:\r<BR>\n" );
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document.write( "log(4,(7-3)) = 1 which becomes:\r<BR>\n" );
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document.write( "log(4,4) = 1\r<BR>\n" );
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document.write( "Use the base conversion formula to get:\r<BR>\n" );
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document.write( "log(4,4) = log(10,4)/log(10,4) which becomes:\r<BR>\n" );
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document.write( "log(4,4) = 1, confirming that the answer is good if I interpreted the problem correctly.\r<BR>\n" );
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document.write( "Your last problem is e^t = 1000.\r<BR>\n" );
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document.write( "To solve this, you need to take the log of both sides of this equation to get:\r<BR>\n" );
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document.write( "log(e^t) = log(1000).\r<BR>\n" );
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document.write( "The base of 10 is implied.\r<BR>\n" );
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document.write( "The formula could also be written as log(10,e^t) = log(10,1000).\r<BR>\n" );
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document.write( "Since, in general, log(x^y) = y*log(x), your equation becomes:\r<BR>\n" );
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document.write( "t*log(e) = log(1000).\r<BR>\n" );
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document.write( "Divide both sides of this equation by log(e) to get:\r<BR>\n" );
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document.write( "t = log(1000)/log(e).\r<BR>\n" );
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document.write( "e is equal to the scientific constant of 2.718281828.\r<BR>\n" );
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document.write( "Use your calculator to solve for t to get:\r<BR>\n" );
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document.write( "t = 6.907755279.\r<BR>\n" );
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document.write( "Substitute in your original equation to get:\r<BR>\n" );
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document.write( "e^6.907755279 = 1000.\r<BR>\n" );
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document.write( "Use your calculator to solve to get:\r<BR>\n" );
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document.write( "1000 = 1000, confirming that the answer is good.\r<BR>\n" );
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document.write( "The formulas that you needed to use to solve these problems are:\r<BR>\n" );
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document.write( "y = log(b,x) if and only if b^y = x.\r<BR>\n" );
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document.write( "That's the basic definition of logarithms.\r<BR>\n" );
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document.write( "Also:\r<BR>\n" );
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document.write( "log(x^y) = y*log(x).\r<BR>\n" );
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document.write( "Also:\r<BR>\n" );
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document.write( "log(b,x) = log(c,x) / log(c,b)\r<BR>\n" );
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document.write( "That's the base conversion formula.\r<BR>\n" );
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document.write( "I think I interpreted the problem correctly, but I may have gotten it wrong.\r<BR>\n" );
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document.write( "Terminology:\r<BR>\n" );
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document.write( "log(b,x) means log of x to the base of b.\r<BR>\n" );
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document.write( "log(x) means log of x to the base of 10.\r<BR>\n" );
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document.write( "Your calculator LOG function solves for logs to the base of 10.\r<BR>\n" );
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document.write( "Your calculator LN function solves for logs to the base of e.\r<BR>\n" );
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document.write( "e is the scientific constant of 2.718281828.\r<BR>\n" );
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document.write( "The basic definition of logarithms as it applies to the base of e is as follows:\r<BR>\n" );
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document.write( "log(b,x) = y if and only if b^y = x\r<BR>\n" );
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document.write( "Applied to a base of e, that becomes:\r<BR>\n" );
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document.write( "log(e,x) = y if and only if e^y = x\r<BR>\n" );
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document.write( "In your last problem, you could also have used the basic definition of logs to solve it.\r<BR>\n" );
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document.write( "That problem was e^t = 1000\r<BR>\n" );
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document.write( "By the basic definition of logs, that is true if and only if:\r<BR>\n" );
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document.write( "ln(1000) = t\r<BR>\n" );
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document.write( "Note that ln(1000) is the same as log(e,1000).\r<BR>\n" );
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document.write( "Since your calculator has an LN function key (or should have), you could have then used your calculator to solve this.\r<BR>\n" );
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document.write( "You would have gotten:\r<BR>\n" );
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document.write( "ln(1000) = 6.907755279\r<BR>\n" );
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document.write( "That's exactly the same answer we got before, as it should be.\r<BR>\n" );
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document.write( "You may be interested in my lessons on logarithms and exponents.\r<BR>\n" );
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document.write( "They can be found at the following links:\r<BR>\n" );
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document.write( " </SPAN>\n" );
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