The number of positive values \(x\) which satisfy the equation \[x=8^{\log_2x} -9^{\log_3x}-4^{\log_2x}+\log_{0.5}{0.25}\] is

  1. 0;

  2. 1;

  3. 2;

  4. 3.

We note that \(\log_{0.5}0.25=\log_{0.5}(0.5)^2=2\).

If \(x>0\) then \[8^{\log_2x}=(2^3)^{\log_2x}=(2^{\log_2x})^3=x^3.\]

Similarly, \[9^{\log_3x}=(3^2)^{\log_3x}=(3^{\log_3x})^2=x^2\] and \[4^{\log_2x}=(2^2)^{\log_2x}=(2^{\log_2x})^2=x^2.\]

So the equation becomes \[x=x^3-x^2-x^2+2,\] that is, \[x^3-2x^2-x+2=0.\]

We can see that \(x=2\) is a solution, so \((x-2)\) is a factor. By inspection, we find \[(x-2)(x^2-1)=0\] and so \[(x-2)(x-1)(x+1)=0.\]

So the roots are \(x=-1\), \(1\) and \(2\).

There are two positive values of \(x\) satisfying the equation so the answer is c.