Question

Log\(_{2}\)(x) + log\(_{2}\)(x-2) = 3

• A. 2
• B. 3
• C. 4
• D. 5

Hint:

Always check your solutions to logarithmic equations by substituting them back into the original equation to ensure the arguments of the logarithms remain positive.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to solve the given logarithmic equation for the variable x. 
Step 2: Key Formula or Approach: 
We will use the following properties of logarithms: 
1. Product Rule: log\(_{b}\)(m) + log\(_{b}\)(n) = log\(_{b}\)(mn) 
2. Conversion to Exponential Form: If log\(_{b}\)(a) = c, then b\(^c\) = a. 
Step 3: Detailed Explanation: 
First, apply the product rule to combine the two log terms: \[ \log_{2}(x(x-2)) = 3 \] Next, convert the logarithmic equation to its exponential form: \[ 2^3 = x(x-2) \] \[ 8 = x^2 - 2x \] Rearrange the equation into a standard quadratic form (ax\(^2\) + bx + c = 0): \[ x^2 - 2x - 8 = 0 \] Factor the quadratic equation: \[ (x-4)(x+2) = 0 \] This gives two possible solutions for x: x = 4 or x = -2. 
However, we must check these solutions against the domain of the original logarithmic expressions. The argument of a logarithm must be positive.
- For log\(_{2}\)(x), we must have x $>$ 0.
- For log\(_{2}\)(x-2), we must have x-2 $>$ 0, which means x > 2.
The solution must satisfy both conditions, so we need x $>$ 2. 
- The solution x = 4 is valid because 4 $>$ 2.
- The solution x = -2 is invalid (extraneous) because it does not satisfy x > 2. 
Step 4: Final Answer: 
The only valid solution to the equation is x = 4.