The number of solutions of `log_(2) (x - 1) = 2 log_(2) (x - 3)` :

Options

A. A)2
B. B)1
C. C)6
D. D)7

Answer

To solve the equation \( \log_{2}(x - 1) = 2 \log_{2}(x - 3) \), we will follow these steps:

### Step 1: Rewrite the equation using logarithmic properties
We can use the property of logarithms that states \( a \log_b(c) = \log_b(c^a) \). Thus, we can rewrite the right-hand side:
\[ \log_{2}(x - 1) = \log_{2}((x - 3)^2) \]

### Step 2: Remove the logarithm by exponentiating both sides
Since the logarithms are equal, we can set the arguments equal to each other:
\[ x - 1 = (x - 3)^2 \]

### Step 3: Expand the right-hand side
Now, we expand the right-hand side:
\[ x - 1 = x^2 - 6x + 9 \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ 0 = x^2 - 6x + 9 - x + 1 \]
\[ 0 = x^2 - 7x + 10 \]

### Step 5: Factor the quadratic equation
Now we can factor the quadratic:
\[ 0 = (x - 2)(x - 5) \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us the solutions:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]

### Step 7: Check the validity of the solutions
We need to check if these solutions are valid in the context of the logarithmic functions:
- For \( x = 2 \):
- \( x - 1 = 2 - 1 = 1 \) (valid)
- \( x - 3 = 2 - 3 = -1 \) (invalid, as logarithm of a negative number is undefined)

- For \( x = 5 \):
- \( x - 1 = 5 - 1 = 4 \) (valid)
- \( x - 3 = 5 - 3 = 2 \) (valid)

Thus, only \( x = 5 \) is a valid solution.

### Conclusion
The number of solutions to the equation \( \log_{2}(x - 1) = 2 \log_{2}(x - 3) \) is **1**.

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