Question

The number of solutions of \[ \log_4 (x - 1) = \log_2 (x - 3) \] is/are:

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

When solving logarithmic equations, always check for extraneous solutions, particularly when dealing with logarithms of negative numbers or zero.

Answer 1

The Correct Option is B

Step 1: Convert logarithms to the same base.
We are given the equation:
\[ \log_4 (x - 1) = \log_2 (x - 3) \]
We can convert \( \log_4 \) to base 2:
\[ \log_4 (x - 1) = \frac{1}{2} \log_2 (x - 1) \]
Thus, the equation becomes:
\[ \frac{1}{2} \log_2 (x - 1) = \log_2 (x - 3) \]

Step 2: Eliminate the fractions.
Multiply both sides of the equation by 2 to eliminate the fraction: \[ \log_2 (x - 1) = 2 \log_2 (x - 3) \]

Step 3: Apply logarithmic properties.
Using the logarithmic property \( a \log_b x = \log_b x^a \), we can rewrite the equation as: \[ \log_2 (x - 1) = \log_2 ((x - 3)^2) \]

Step 4: Solve the equation.
Since the logarithms are now equal, the arguments must be equal as well: \[ x - 1 = (x - 3)^2 \] Expanding the square: \[ x - 1 = x^2 - 6x + 9 \]

Step 5: Rearrange the equation.
Rearrange the equation to bring all terms to one side: \[ 0 = x^2 - 7x + 10 \] Factor the quadratic: \[ 0 = (x - 5)(x - 2) \]

Step 6: Find the solutions.
Thus, the solutions are: \[ x = 5 \quad \text{or} \quad x = 2 \] We must check if these solutions satisfy the original logarithmic equation. For \( x = 5 \): \[ \log_4 (5 - 1) = \log_2 (5 - 3) \implies \log_4 4 = \log_2 2 \implies 1 = 1 \quad \text{(valid solution)} \] For \( x = 2 \): \[ \log_4 (2 - 1) = \log_2 (2 - 3) \implies \log_4 1 = \log_2 (-1) \quad \text{(invalid solution as logarithms of negative numbers are undefined)} \]

Step 7: Conclusion.
Thus, the only valid solution is \( x = 5 \), so the number of solutions is 1.