document.write( "Question 202723: This is the new one:
\n" ); document.write( "log(25,1/5.log(3,2-log(0.5,x)))=-1/2 \n" ); document.write( "
Algebra.Com's Answer #152901 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
If the equation is
\n" ); document.write( "\"log%2825%2C%28%281%2F5%29log%283%2C%282-log%280.5%2Cx%29%29%29%29%29=-1%2F2\"
\n" ); document.write( "then the solution, which I call \"peel the onion\", follows. If this equation is not correct, try again this time:
  • Start your equation with three left braces (shift [) and finish the equation with three right braces (shift ])
  • Use more parentheses to make sure everything is properly grouped
    \n" ); document.write( "Although not required, we are going to use an additional variable, z, to help make what we are doing easier to understand.

    \n" ); document.write( "First we'll make \"z+=+%281%2F5%29log%283%2C%282-log%280.5%2Cx%29%29%29\"
    \n" ); document.write( "Substituting z into the original equation we get:
    \n" ); document.write( "\"log%2825%2C+z%29+=+-1%2F2\"
    \n" ); document.write( "Rewriting this in exponential form we get:
    \n" ); document.write( "\"z+=+25%5E%28-1%2F2%29+=+1%2F%2825%5E%281%2F2%29%29+=+1%2F%28sqrt%2825%29%29+=+1%2F5\"
    \n" ); document.write( "So z = 1/5. Now we will substitute back the original expression for z:
    \n" ); document.write( "\"%281%2F5%29log%283%2C%282-log%280.5%2Cx%29%29%29+=+1%2F5\"
    \n" ); document.write( "Multiplying both sides by 5 we get:
    \n" ); document.write( "\"log%283%2C%282-log%280.5%2Cx%29%29%29+=+1\"
    \n" ); document.write( "Now we will reuse z (or use another variable if this make you uncomfortable). Let z = the argument of the outer log function.
    \n" ); document.write( "\"z+=+2-log%280.5%2Cx%29%29\"
    \n" ); document.write( "Substituting:
    \n" ); document.write( "\"log%283%2C+z%29+=+1\"
    \n" ); document.write( "Rewriting in exponential form:
    \n" ); document.write( "\"z+=+3%5E1+=+3\"
    \n" ); document.write( "Substituting back for z:
    \n" ); document.write( "\"2-log%280.5%2Cx%29+=+3\"
    \n" ); document.write( "Subtracting 2 from both sides we get
    \n" ); document.write( "\"-log%280.5%2C+x%29+=+1\"
    \n" ); document.write( "Dividing (or multiplying) both sides by -1 we get
    \n" ); document.write( "\"log%280.5%2C+x%29+=+-1\"
    \n" ); document.write( "Rewriting in exponential form we get
    \n" ); document.write( "\"x+=+%280.5%29%5E%28-1%29+=+%281%2F2%29%5E%28-1%29+=+2%2F1+=+2\"
    \n" ); document.write( "So our solution is: x = 2. I call this \"peel the onion\" because the x was buried inside a log which was buried inside anothe log which was buried inside a 3rd log. So I \"peeled away\" from the outside until x was by itself. \n" ); document.write( "
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