If the equation ln `(x^(2) +5x ) -ln (x+a +3)=0` has exactly one solution for x, then possible integral value of a is:

Options

A. `-3`
B. `-1`
C. 0
D. 2

Answer

To solve the equation \( \ln(x^2 + 5x) - \ln(x + a + 3) = 0 \) for the condition that it has exactly one solution for \( x \), we can follow these steps:

### Step 1: Combine the logarithmic expressions
Using the property of logarithms that states \( \ln A - \ln B = \ln \frac{A}{B} \), we can rewrite the equation as:
\[ \ln\left(\frac{x^2 + 5x}{x + a + 3}\right) = 0 \]

### Step 2: Exponentiate both sides
Exponentiating both sides gives us:
\[ \frac{x^2 + 5x}{x + a + 3} = 1 \]

### Step 3: Clear the fraction
Multiplying both sides by \( x + a + 3 \) (assuming \( x + a + 3 \neq 0 \)):
\[ x^2 + 5x = x + a + 3 \]

### Step 4: Rearrange the equation
Rearranging the equation gives us:
\[ x^2 + 5x - x - a - 3 = 0 \]
which simplifies to:
\[ x^2 + 4x - (a + 3) = 0 \]

### Step 5: Determine the condition for exactly one solution
For the quadratic equation \( x^2 + 4x - (a + 3) = 0 \) to have exactly one solution, the discriminant must be zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Here, \( a = 1 \), \( b = 4 \), and \( c = -(a + 3) \). Thus, we have:
\[ D = 4^2 - 4 \cdot 1 \cdot (-(a + 3)) = 16 + 4(a + 3) \]
Setting the discriminant equal to zero for one solution:
\[ 16 + 4(a + 3) = 0 \]

### Step 6: Solve for \( a \)
Simplifying the equation:
\[ 16 + 4a + 12 = 0 \]
\[ 4a + 28 = 0 \]
\[ 4a = -28 \]
\[ a = -7 \]

### Step 7: Check for integral values
Since we are looking for possible integral values of \( a \), we find that \( a = -7 \) is the only solution that satisfies the condition of the quadratic having exactly one solution.

### Final Answer
The possible integral value of \( a \) is:
\[ \boxed{-7} \]