Logarithms

Consider:

Let:

Thus:

Substitute and simplify:

Multiplying through by gives:

Solving this quadratic equation:

Since:

Thus, the corresponding values of are:

Number of solutions = 2.

Step 1: Write the conditions for logarithmic function.
For to be defined: Here, Step 2: Solve .
Roots: Since the coefficient of is positive: Step 3: Solve .
Roots: Hence, Step 4: Exclude .
Roots: These values must be excluded from the domain. Step 5: Combine all conditions.
Intersecting all valid intervals: Thus, Step 6: Compute the required value.