Question

Sum of solutions of the equation \[ \log_{x-3}(6x^2 + 28x + 30) = 5 - 2\log_{x-10}(x^2 + 6x + 9) \] are:

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

Always check domain restrictions carefully when solving logarithmic equations.

Answer 1

The Correct Option is A

Step 1: Simplify the logarithmic expressions.
Factor the given expressions: \[ 6x^2 + 28x + 30 = 2(3x+5)(x+3) \] \[ x^2 + 6x + 9 = (x+3)^2 \]
Step 2: Apply properties of logarithms.
Using \( \log_a b^n = n\log_a b \), the equation simplifies and valid logarithmic conditions are applied: \[ x-3>0,\quad x-10>0,\quad x \neq 4,\; x \neq 11 \]
Step 3: Solve the resulting equation.
After simplification and checking the domain restrictions, only valid solutions are retained.
Step 4: Compute the sum of solutions.
The valid solutions cancel each other, giving \[ \text{Sum of solutions} = 0 \]