document.write( "Question 246026: find the domain of the function: f(x)=log(base 7)(2-(x+5/x-6)i need to show the work \n" ); document.write( "
Algebra.Com's Answer #179716 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"f%28x%29=log%287%2C+%282-%28x%2B5%29%2F%28x-6%29%29%29\"

\n" ); document.write( "There are two issues to address here:
  • Denominators may not be zero.
  • Arguments of logarithms, no matter what the base, must not be zero or negative.

\n" ); document.write( "Denominator.
\n" ); document.write( "Let's find what x value makes the denominator zero:
\n" ); document.write( "x - 6 = 0
\n" ); document.write( "Add 6 to both sides:
\n" ); document.write( "x = 6
\n" ); document.write( "This is a value we cannot allow for x. It must be excluded from the domain.

\n" ); document.write( "Argument of the logarithm.
\n" ); document.write( "We want a positive argument:
\n" ); document.write( "\"2-%28x%2B5%29%2F%28x-6%29+%3E+0\"
\n" ); document.write( "There are different ways to solve this. One way is to subtract the terms on the left. Of course we need common denominators first:
\n" ); document.write( "\"2%28%28x-6%29%2F%28x-6%29%29-%28x%2B5%29%2F%28x-6%29+%3E+0\"
\n" ); document.write( "\"%282x-12%29%2F%28x-6%29-%28x%2B5%29%2F%28x-6%29+%3E+0\"
\n" ); document.write( "Subtract (Be careful when subtracting!) :
\n" ); document.write( "\"%28x+-17%29%2F%28x-6%29+%3E+0\"
\n" ); document.write( "This says we have a positive fraction. When do we get positive fractions? Answer: When the numerator and denominator are both positive or when they are both negative. This idea, which you probably knew well, can be expressed in the algebraic form:
\n" ); document.write( "(\"x-17+%3E+0\" and \"+x-6+%3E+0\") or (\"x-17+%3C+0\" and \"+x-6+%3C+0\")
\n" ); document.write( "(Look at this and see if you can understand how this says \"both positive or both negative\".)
\n" ); document.write( "Now we solve this compound inequality.
\n" ); document.write( "(\"x+%3E+17\" and \"+x+%3E+6\") or (\"x+%3C+17\" and \"+x+%3C+6\")
\n" ); document.write( "In the first pair, in order for x to be greater than 17 and 6, it would have to be greater than 17. In the last pair, in order for x to be less than 17 and 6, it would have to be less than 6. So
\n" ); document.write( "(\"x+%3E+17\" and \"+x+%3E+6\") or (\"x+%3C+17\" and \"+x+%3C+6\")
\n" ); document.write( "simplifies to
\n" ); document.write( "\"x+%3E+17\" or \"+x+%3C+6\"

\n" ); document.write( "Since \"x+%3E+17\" or \"+x+%3C+6\" excludes the number which makes the denominator zero, this is our domain.
\n" ); document.write( "
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