Q. Select all the solutions of the equation (2log x = log(5x – 4)).

Problem

Solve the equation \(2\log x = \log(5x – 4)\) and select all solutions from the list: a. \(x=1\), b. \(x=4\), c. \(x=5\), d. \(x=0\), e. \(x=-1\).

1. Domain (where both logarithms are defined):

   For \(\log x\) we need \(x>0\). For \(\log(5x-4)\) we need \(5x-4>0\), which is \(x>\tfrac{4}{5}\). Both conditions together give the domain \(x>\tfrac{4}{5}\).

2. Use logarithm property to combine the left side:

   The property \(n\log a = \log(a^n)\) gives \(2\log x = \log(x^2)\). So the equation becomes

   \(\log(x^2) = \log(5x-4)\).

3. Equate arguments (same base logs):

   Since the logs (same base) are equal and the expressions are in the domain, the arguments are equal:

   \(x^2 = 5x – 4\).

4. Solve the resulting quadratic:

   Rearrange:

   \(x^2 – 5x + 4 = 0\).

   Factor:

   \((x-1)(x-4) = 0\).

   Thus the algebraic solutions are \(x=1\) and \(x=4\).

5. Check against the domain:

   Both \(x=1\) and \(x=4\) satisfy \(x>\tfrac{4}{5}\), so both are valid solutions of the original equation.

6. Conclusion (which choices are correct):

   The solutions are \(x=1\) and \(x=4\). Therefore the correct choices are a and b.